It is one of the many great debates in the Spanish language, and is really a result of two quite different uses of the conjugation:
- Iría a clase si no lloviese/lloviera.
Clearly indicates something that is not occurring, will not occur, and has not occurred. Thus, one could make a solid argument for a non-indicative mood.
On the other hand, you also have statements like this:
- Dijo que iría a clase a las ocho.
While the event had not yet occurred at the moment that dijo occurs, it is presumed that it will happen and in fact is fully interchangeable with the definitively indicative phrase iba a ir. If it's 100% interchangeable with the indicative, it must also be indicative, the thought would go.
Still others look to the syntactical nature. Subjunctive can't be used generally without a main clause, most grammarians will point out, but the restriction isn't absolute: exhortative subjunctive is still alive and well in the form of usted and nosotros commands, and all negative commands, and also in formal Spanish. Some grammarians will argue that in reality, such constructions have an implicit indicative clause like "(quiero que) no me hables", but I'm less convinced, as in subordinate clauses proclisis is mandatory, but "hábleme" is perfectly grammatical with enclisis.
Some grammarians, on similar lines, will ague for our against conditional's status based on its ability to be used in the main clause, but others will say that in such cases, there's really just an implicit clause in the subjunctive, e.g., "yo iría (si fuese así)". But, the subjunctive is seen as being the dependent that attaches to an independent, so despite requiring an additional phrase (which would make it seem dependent/subordinate) it gets analyzed as independent.
Another analysis combines both the conditional and the future into present-y/past-y forms of a different mood: the potential because they express the capacity for doing things now or down the line and because they form etymologically from the same verbal periphrasis, just in different tenses (inf. + present of haber = future; inf. + old imperfect of haber = conditional). I personally subscribe to this view.
All of this is to say that the conditional occupies a very odd place in Spanish grammar, and defining the conditional as one thing or another will be heavily dependent on how you define a number of other parts of the Spanish language. To wit, here is how the Royal Spanish Academy introduces the conditional:
23.15a Es tradicional la polémica en torno a si el condicional (cantaría) es un tiempo del indicativo o un modo verbal. Se suele aceptar hoy la primera opción […] Aun así, es oportuno recordar que el condicional no expresa solo situaciones orientadas en relación con el momento del habla o con otro punto que se tome como eje, sino también contenidos no factuales (§ 25.1c) supeditados a situaciones hipotéticas, lo que se deduce de su orientación prospectiva. Se ha señalado que el hecho de que las oraciones formadas con este tiempo sean proposiciones supeditadas a otras impide otorgarles con nitidez un valor de verdad, más claramente incluso que a las construidas en futuro.
So while the Gramática treats it as being a part of the indicative, it also recognizes that that is not an absolute position (in other words, "let's agree to disagree, but this is the terminology we're using because we need to settle on one for a 2000 page book, and we'll point out problems with it as they come up").