In this case, the verb haver has the semantic meaning of time that has gone through.
The analysis in this case is a verbal / clausal analysis, considering the adverbial a finite clause, with the verb haver as the main verb.
 

ADVERBIAL CLAUSE

ADVL:fcl
=MV:v-fin    há
=ACC:np
==>N:num    dois
==H:n    anos
 

The same semantics of the verb haver may occur in other type of constructions like  Há <time length> que....as in the example below:

C52-1:  Há muito tempo que tenho uma estranha relação afectuosa com esta ilha.

The syntactic analysis of sentences of this structure: <Há / havia> <expression of time> que,  involves complex issues.  The first one connected to the morphological category of "há", recognised as a preposition in the VISL system.
The other question is related to the analysis of the (hypothetical) clause: que tenho uma estranha relação afectuosa com esta ilha.
Let's start by looking at the parser's output of this sentence:

há [há] <*> PRP @ADVL                 [haver] <*> <vUK> <ink> <fmc> V PR 3S IND VFIN @FMV

muito=tempo [muito=tempo] <am> <dur> <temp> N M S @P<

que [que] <rel> <que-n> SPEC M/F S/P @ACC> @#FS-N<

tenho [ter] <vt> <ink> V PR 1S IND VFIN @FMV

uma [um] <arti> DET F S @>N

estranha [estranho] <jh> ADJ F S @>N

relação [relação] <+entre> N F S @<ACC

estreita [estreito] <n> ADJ F S @N<

com [com] PRP @<ADVL

esta [este] <dem> DET F S @>N

ilha [ilha] <top> N F S @P<
.

Both prepositional and verbal readings of the verb haver are considered. However, the sentence analysis is based on the prepositional reading (muito tempo: @P< but there's no @<ACC alternative).
The real problem arises when it comes to analyse que as a relative pronoun initiating a relative clause.

Let's then look at the possibilities for the analysis of que:
 

1. pronominal relative pronoun

    If so, then it may refer to the previous noun (as considered by the parser) or noun phrase or sentence (@S<)- sentence apposition.
However, notice the ungrammaticality of the following sentences:

 (1a) *O que tenho uma estranha relação afectuosa com esta ilha? Muito tempo.
 (1b) * Há muito tempo. Muito tempo tenho uma estranha relação afectuosa com esta ilha.

2. Subordinating conjunction: "que "

Subordinating conjunctions initiate a subordinated clause, which must have a syntactic function in relation to the main clause, like "Ela disse que tinha uma estranha relação afectuosa com a ilha", que tinha uma estranha relação afectuosa com a ilha is the complement of the verb dizer, main verb of the main clause Ela disse.
Bearing in mind the syntax of the verb haver (C12-3) and the uniqueness principle of the direct object:

    2.1. "Há muito tempo" would be the main clause (and the verbal reading of haver the only possible reading);
    2.2. "que tenho uma estranha relação afectuosa com esta ilha", a subordinated clause.

há [haver]<*> <vUK> <ink> <fmc> V PR 3S IND VFIN @FMV
muito=tempo [muito=tempo] @<ACC
que [que]    KS @SUB  @#FS-< ?
tenho [ter] <vt> <ink> V PR 1S IND VFIN @FMV
uma [um] <arti> DET F S @>N
estranha [estranho] <jh> ADJ F S @>N
relação [relação] <+entre> N F S @<ACC
estreita [estreito] <n> ADJ F S @N<
com [com] PRP @<ADVL
esta [este] <dem> DET F S @>N
ilha [ilha] <top> N F S @P<
.

Which external function would the subordinate clause hold? It cannot be subject (@SUBJ) as haver is an impersonal verb:

(2.2.a) *Que tenho uma estranha relação afectuosa com esta ilha há muito tempo.

The argument that the verb obligatorily selects is the direct object (@ACC) which follows the uniqueness principle (a sentence can only present one direct object). That leads to hypothesise about what the direct object of the main verb is: muito tempo or the "subordinate" clause, and it is not clear. Pronominalising the direct object(s):

(2.2.b) (?) Há-o que tenho uma estranha relação afectuosa com esta ilha.
(2.2.c) (?) Há muito tempo isto   but    * Há isto muito tempo, (isto standing for the "que-clause")

which indicates that "muito tempo" alone cannot function as an @ADVL (like other time adverbials like ontem, hoje, and so on) and, therefore, cannot be separated from the verb haver. This might be a good argument to consider muito tempo the argument selected by haver.

Furthermore, if the "que-clause" was an object clause, being que the conjunction, then que could not be omitted which does not apply to this particular example:

(2.2.d) Tenho uma estranha relação afectuosa com esta ilha há muito tempo.

If "há muito tempo" is mantained in the beginning of the sentence, oddity arises:

(2.2.e)  (?) Há muito tempo tenho uma estranha relação afectuosa com esta ilha

but

(2.2.f) Há muito tempo tinha / tive uma relação afectuosa com esta ilha.

These changes in the verb carry a difference in semantics. Apparently, que does not hold a relevant syntactic function to the sentence as a whole, being a dummy operator, like in the following sentences:

(S1) Que tenhas um bom-dia!
(S2) Desde 1990 que estava na mesa a reformulação das "secretas". (source CETEMPúblico)

In (S2), que is not obligatory:  Desde 1990 estava na mesa a reformulação das "secretas".
 

Despite that fact, que seems to interfere with meaning.  Let's examine some examples:

(Q1)  De facto, há muitos anos que há quem proponha a utilização de pequenas bombas nucleares em certas grandes obras de engenharia como terraplanagens de montanhas ou abertura de albufeiras.(source CETEMPúblico)

(Q2)  Isso poderia revestir-se de particular gravidade se o ataque ocorresse dentro de Angola, onde há poucos dias Ø  dois padres e uma freira foram mortos num ataque de «bandidos» . (source CETEMPúblico)

The difference between the two sentences is in the time that is referred. That is, while in (Q1), there is a clear continuity, in (Q2) that continuity does not exist for two reasons:
        1. the verb tense is fixed:  *...., onde há poucos dias dois padres e uma freira são mortos
                                                        (?) ..., onde há poucos dias dois padres e uma freira eram mortos (possible, if describing the act of murdering)

           2. The verb itself (matar) does not allow that continuity.

Therefore, in the (Q1) the time reference is a period of time (from a long time ago until now, and possibly is not finished), while in (Q2) the time reference is a specific moment in time (some days ago).
The above can be schematised in the following diagrams:

(Q1)
                                                                 year x-n        "Há muitos anos que..."                             year x
                                ---------------|---------------------------|------------>
                                                                   "proposta"                                                                     "moment of speech"           t
 

(Q2)

                                                                    day x-n                                                                                day x
                                ---------------|---------------------------|--------------->
                                                             "....foram mortos"                                                          "moment of speech"               t
 
 

The fact that que can be  a mark for two different time references is corroborated if the questions "Quando?" (When?) and "Desde quando?" (since when?) are asked:

(Q'1) Desde quando há quem proponha a utilização de bombas nucleares.....?
            Há muito tempo que há quem proponha.....

         but

            *Quando há quem proponha a utilização de bombas nucleares ...?
            *Há muito tempo que há quem proponha....

(Q'2) Quando foram mortos dois padres e uma freira?
            Há poucos dias foram mortos....

         but

            *Desde quando foram mortos dois padres e uma freira?
            *Há poucos dias foram mortos.

Furthermore, if we invert the order, one can realise that the meaning does not change in (Q2) but it might become ambiguous in (Q2):

(Q''1) Há quem proponha a utilização de bombas nucleares há muito tempo.

The verb in the present tense makes the non-continuity reading impossible. But if the tense is changed, the semantic ambiguity arises:

(T1) Há muito tempo que havia quem propusesse a utilização de bombas nucleares... (continuity in a period of time)

(T2) Havia quem propusesse a utilização de bombas nucleares (...) há muito tempo (unclear: both questions "Quando?" and "Desde quando?" can be applied).

(Q''2) ....dois padres e uma freira foram mortos há poucos dias. (no change in meaning).
 
 

3. Adverbial relative pronoun

Having discarded the object reading of the 'que-clause', one might consider another analysis:  the 'que-clause' as an adverbial relative clause, being the preposition (durante, for instance) elliptic:

(2.2.g) Há muito tempo (durante) o qual (=que) tenho uma estranha relação afectuosa com esta ilha.

The correspondent syntactic tree is the following:

STA:fcl
P:v-fin('haver' PR 3S IND) Há
ACC:np('muito_tempo' M S) muito_tempo
N<:fcl
=ADVL:adv('que' <rel>) que
=P:v-fin('ter' PR 1S IND) tenho
=ACC:np
==>N:art('um' <arti> F S) uma
==>N:adj('estranho' F S) estranha
==H:n('relação' F S) relação
==N<:adj('afectuoso' F S) afectuosa
==N<:pp
===H:prp('com') com
===P<:np
====>N:pron-det('este' <dem> F S) esta
====H:n('ilha' F S) ilha
.

Taking  the continuity aspect of (T1), the pronominal relative clause reading seems to fit quite acceptably, at least semantically:
 

(T1') Há muito tempo (durante) o qual (=que) havia quem propusesse a utilização de bombas nucleares.

Bearing in mind the (time) graphs presented above, this analysis points to them as it indicates a period in which something happened. Reinforcing this idea, is the fact that when a single moment in time, not a continuous period, is at stake, que is not an obligatory element in the sentence.
 Therefore, the proposed analysis takes há / havia as the finite main verb, at the clausal level, the expression of time the ACC  and the clause initiated by que, an adverbial relative clause.
 
 
 

Parser's default:

There seems to be a clear difference in meaning by the use or not of 'que' and one could argue that its use indicates that somehow there is a topic construction- the @ADVL "Há muito tempo " is the known information and the rest of the sentence the new information-  and, for that reason it ought to be tagged as @FOC.  This is in fact the default analysis:

ADVL:fcl
=P:v-fin('haver')    Há / Havia
=ACC:np ('muito_tempo')    muito_tempo
FOC:adv('que')    que
 

Despite the fact that both constructions of há / havia semantically related to time (with or without 'que') hold the adverbial clause reading, historically (and internally, still) the parser opts the disambiguationally and technically simplest solution: há / havia as prepositions as:

1. adverbials are typically prepositional phrases, and a preposition typically the head;
2. functionally, it presents the same behaviour: invariable word, requiring a complement.
 

However, there is no tradition or of having a form of a verb (haver) classified as a preposition and so the pp-construction is filtered into the above explained ADVL:fcl.