I can answer the second prong of this question. Same-sex marriage didn't become a legal possibility in New York until the 2000s, first with the recognition (for limited government purposes) of same-sex marriages conducted in other jurisdictions where that was a valid legal possibility; then with recognition of out-of-state same-sex marriages more generally; then with full legalization in 2011. Point being, Elaine is heading to a lesbian wedding in New York, and New York did not permit same-sex weddings at the time. So no, it would not be valid.

In the abstract, you could perhaps argue that same-sex marriage was not explicitly illegal in New York at the time, and thus existed in a gray area between being undoubtedly legal and undoubtedly illegal, because I'm not aware that any New York statute or binding appellate precedent bothered to define marriage as between a man and a woman at that time. But this is a weak argument. Every court in the country to have considered whether same-sex marriage could be legal had issued a firm no, and while I don't happen to know of any appellate New York precedent, trial courts found the issue straightforward. Anonymous v. Anonymous is an example with some unusual facts, but the upshot is that in 1971 this New York trial court (i.e. not one that was creating appellate precedent binding lower courts) could simply state that marriage was between a man and a woman, cite a single supporting authority from ten years earlier, and leave it at that. Indeed, the court did not find that this same-sex marriage was a null marriage - which could imply same-sex marriage could be valid if not for some defect present here, like the deception present in Anonymous - but made a more aggressive holding that, because same-sex marriage is an inherently invalid concept, the legal ceremony itself was a nullity and gave rise to nothing, not even a null marriage. (This may seem like an empty distinction, but it does matter.) It may also be worth noting that this relatively early example of same-sex marriage in the law didn't come from LGBT litigants seeking to force the state to recognize same-sex marriages; quite the opposite, it came from a servicemember trying to get out of marital status. In short, it can't credibly be said that a lesbian wedding taking place in 1992, when the episode is set, might potentially have been legal or existed in a gray area of legality: it would have been clear that New York law did not recognize such a marriage. So no, there was no legal weight behind the marriage in the episode.

I have two qualifiers to add to this. First, it's not really about marriage, but it may be interesting to note that in 1989, New York's highest court held that, when one partner in a same-sex couple living together in a rent-controlled apartment died, the other partner was to be considered family of the deceased, such that they could continue to live in the rent-controlled apartment. I'm not qualified to say where this compared to other states on the issue of LGBT rights, but Braschi is viewed as a landmark case for recognizing that the bonds of affection and care between same-sex couples qualified as family ties just as those between opposite-sex couples did - and in what was then the second most populous state, too. Residency in rent-controlled apartments may seem like pretty small potatoes compared to the simpler and broader issue of marriage, but Braschi is nonetheless an important step forward in the law, and a significant data point to show that the legal prospects of LGBT issues were already starting to improve by the late 1980s, if only slowly.

Second, this answer only deals with marriage as a legal institution. And in that limited sense, it also answers the first half of your question: same-sex marriages in the early 90s were nonexistent, not common, because no jurisdiction allowed them under the law. That feels a little glib. I don't think that's what you're asking about, and it's certainly true that marriage is broader than how the law defines it. I have no idea how common it was in the early 90s for same-sex couples to hold ceremonies affirming their bonds of love and commitment, which were valid in many senses whatever the law or various mainstream religions might have had to say about them, so someone else would have to answer that. But I can say, to end on a bit of a happier note - and notwithstanding that, as I noted above, there's little argument that same-sex marriage was clearly not legally valid under the laws of New York at the time - there was plenty of room for same-sex couples to live as married in some senses, under the right circumstances. Nothing about the ceremony Elaine's friends were holding, or about a same-sex couple living as married, was legally forbidden such that someone could call the cops on them. (New York's sodomy statute had been struck down by the state's highest court in 1980.) So while Elaine's friends couldn't have gotten a marriage license, or forced the state to extend them the tax incentives afforded to opposite-sex couples, nothing at all was stopping the couple from living as married and treating each other as spouses, nor stopping the couple's friends and associates from doing the same if they chose to. If they lived in tolerant enough circumstances, they could have done more than one might think to live as a married couple.