The sentence is not referring to any time past, present of future. It's just referring to an imaginary condition which has never existed and seemingly will never exist. Still, the sentence and other sentences of this type are said and spoken. So what can we say about their nature? Which tense are they, what type are they? Their clauses, etc.
Were is the plural past tense form of be, used here in a counterfactual conditional idiom construction that is given various names, including "subjunctive", which often apply to other European languages, though not to English.
In fact, however, tense is not what you need to know here. Tense only has to do with time -- past (was/were) and present (am/is/are) are the only tenses in English -- and the important thing about this construction is not when it occurs, but whether it occurs at all. And it doesn't; nothing happens. That's what counterfactual means.
So, in essence, there is no real time reference involved. Especially since the second clause uses the modal auxiliary could. Modals are defective verbs and are not inflected; therefore they can be said either to be always in the present tense, or to have no tense at all, depending on whether you require the first verb in an auxiliary chain to have a (perhaps inaudible) tense marker; take your pick.
It's true that there are certain uses of modals that retain some of their original preterite morphology, e.g, present can and preterite could in
- When I was 25, I could do 100 pullups; now I can only do 99.
But there's no "past" at all in
- I could do that right now.
- Would you like to dance?
- You must come visit me soon.
which are formed from historically preterite modals (could, might, would, should, must) instead of historically present ones (can, may, will, shall).
They're all idiomatic now. It's a mistake to expect any consistent semantic or grammatical regularity from modals, especially if there are negatives lurking about.
The verbs in this sentence are vestiges of a subjunctive used in English to indicate conditions contrary to fact. We indicate this usage by using the plural form, where it exists, of the past tense.
Were is the plural past tense of to be. Could is the past tense of can (no singular/plural distinction exists).
Example: I have gone to the store to buy milk. If I had gone to the store to buy milk...
We have to distinguish between the name of a tense, eg conditional, and its use.
Present tense can be used to refer to the future time.
"Conditional" as name means this tense/mood mostly occurs in if-clauses. But that does not cover all uses. - I tried but the window would not open, it was jammed. - Here we have a would-form that does not describe an unreal condition.
"Were" in "If I were a bird" is grammatically past subjunctive, but it does not refer to a past event. It expresses an unreal/hypothetical condition referring to present or future time.
So the name of a tense and its use can be very different.
Yes, there are tenses on subjunctive moods.
The tense of "If I were a bird, I would fly to you" is present.
If I write this again in indicative mood: I AM NOT a bird and I CANNOT fly to you.
You also can use past subjunctive mood. "If I had been a bird, I would have flown to you." = (indicative mood) I WAS not a bird and I COULD NOT fly to you.
I would say it is past. In “If I were a bird", the 'were' is the past subjunctive of 'to be'. If it were the present, the sentence would be, “If I be a bird", which is probably even more obsolete than the past form.
The Oxford English Dictionary lists 'were' as the past subjunctive in the 1st, 2nd and 3rd singular and plural. It lists 'be' as the present subjunctive in the 1st, 2nd and 3rd singular and plural. Despite the meaning, it isn't because of the 'were' itself that makes it somewhat irrealis; it is the use of the subjunctive itself, which has two forms