Disclaimer: I am taking advantage of living in the year of 2022 to ask a question about what led to something that happened in 2002, but I also ask about events that happened after that year. Feel free to ignore that part of the question due to the 4th rule of the sub, or to delete this post and tell me to rewrite the title.
In the original version of All The Things She Said (Я сошла с ума), the song begins with "Я сошла с ума (x2) / мне нужна она! (x3)", which can be translated as "I'm crazy (x2) / I need her! (x3)". In the West, however, the lyrics were rewritten as "All the things she said (x2) / runnin' through my head! (x2)".
In the original version of Malchik Gay, "Мальчик-гей (x2) / Будь со мной понаглей / От стыда не красней / Мальчик-гей (x2) / Положи на друзей / Мальчик-гей (x2) / От меня охуей / Мальчик-гей, гей" can be somewhat translated as "Gay boy, (x2) / Be a little naughty with me / Don't blush with shame / Gay boy (x2) / Leave your partner / Gay boy (x2) / Fuck me / Gay boy, gay". In the West, not only this part was rewritten as repetitions of "Malchik gay" and the bridge is sung in Russian, any reference to the gender of this person who seems to have a boyfriend was hidden somehow, including in the name of the song which was left untranslated.
Not Gonna Get Us (Нас не догонят) has very clear queer themes that can be easily interpreted even in the English version of the song, and became one of the most popular songs in the album. Even if the themes can be seen through the songs even in English, it is easily seen (or listened) that in the original Russian lyrics those themes were much more prevalent, and even explicit at moments.
All of this taken into account, the album was released in 2002, ten years after the foundation of the LDRP and one year after the foundation of United Russia, and one year after the foundation of the Eurasia Movement, not to mention Putin becoming the president in 1999 and winning the 2000 elections with more than half of the votes, among other important moments for Russian conservatism in general, from the more acceptable spheres to the most extreme ones.
How did Russia, a country known for it's conservative values, for the prevalence of Orthodox Christianity, for it's anti-queer policies, was able to produce one of the biggest queer cultural symbols of that moment in history? t.A.T.u went to become an international success, and is still being listened by a large number of people, to the point of Трибьют t.A.T.u. "200 по встречной" being released just last year with covers done by popular Russian artists including bands like Molchat Doma and, still today, the band is remembered as a strong cultural icon even if it's members later admitted to not being queer at all and doing it all just for the marketing, which leads to the thought: They not only created a queer cultural icon in literally Russia. They could do it while not being queer. In Russia.
Do we have a clear picture of how Russian society "allowed" this to happen? Why/how was this doable? Was the queer scene in 90s/00s Russia this strong to the point of creating mainstream icons because it was maybe even more accepted and profitable than it appeared to be in the West, if I think about what happened to the lyrics, and if it did, what happened for it to vanish like dust like it never happened, losing power to the point of the Gay Propaganda Law being signed in 2013, forcing the shift from mainstream acts to counterculture icons like Pussy Riot?
As a short question: What situation led to this internationally important moment of Russian queer history amidst the growth of their notorious conservatism, and how did things got so bad since then?
I don't think you'll find many resources to help you with a clear answer to this question, as it's more to do with a complex cultural moment that can't really be explained by pointing to some specific event or situation.
That said, I do want to comment on the framing of your questions because it's really not that uncommon or surprising for a deeply homophobic society to have some popular queer outliers, particularly in the entertainment industry, where they're often viewed as 'non-threatening'. Cultures aren't monoliths, and historically, entertainers have often been 'allowed' to express things they wouldn't otherwise condone for the average citizen. The UK in the 80's was arguably just as homophobic, but figures like Boy George were still wildly popular. But I think the most important thing is that t.A.T.u. never really positioned themselves as LGBT rights advocates or as dissidents and they were never taken seriously as actually queer. As far as I can ascertain, neither band member has ever even publicly supported gay marriage or criticized Putin over his human rights record. Later, after western criticism over their fake lesbianism, t.A.T.u. made some vague gestures like performing at some Pride events, but this was also marred by extremely homophobic comments made by Julia Volkova who basically said that gay men were going against God and were little better than murderers.
The way they were marketed as 'lesbians' was more intended to evoke the porn category and as a male fantasy than actual queer women, so I don't think them being lesbians was ever taken seriously by the mainstream audiences in Russia. Keep in mind that Lena Katina and Julia Volkova were 15 years old(!!!) when t.A.T.u. was formed and only 18 when "All the Things She Said" came out. So them being marketed as lesbians was an addition to the 'edgy,' 'forbidden fruit' image alongside them being underage girls in short shirts.
Here is a quote from a journal article that analyzes media discourse about t.A.T.u. at the time, describing Ivan Shapovalov, the executive behind the band:
"Shapovalov claims that when he first saw Volkova and Katina perform together and sensed the homoerotic energy between them, he decided on the spot to create an ‘underage sex project’, a girl group whose image would depend heavily on illicit sexuality (‘‘‘Lesbian’’ Russian act tops U.K. charts’ 2003). ‘People visit pornographic sites above all others’, Shapovalov has said in explaining his motive. ‘I analysed it and found 90% of people using the Internet go to porno sites first, and of these nine in 10 are looking for underage entertainment. This means there is big interest as well as some dissatisfaction – their needs are not being met’ (Bruce 2003)."
As an Eastern European myself, the main thing I remember when t.A.T.u. became popular in my home country was men making dirty jokes about them where it was clear they didn't see them as anything more than a sexy fantasy, and certainly not a political statement about human rights.
The interpretation that they were intended as a gay rights symbol seems to have mostly come from Western media. Another quote from the article:
"‘It is really funny that the Tatu campaign in the West goes in a very different way to how it went in Russia’, says the Russian music journalist, Artemy Troitsky in an interview with journalist Neil McCormick. ‘In Russia, this whole lesbian thing has never been taken seriously. When they’ve been interviewed on talk shows and asked, ‘‘Are you really lesbians?’’, they’ve said, ‘‘No, it’s a trick, we have boyfriends, we are normal girls, we do this for image’’. And thesame thing for pedophilia. In Russia this whole issue of dirty old men fancying Tatu intheir school uniforms never even broke out. [In Russia], they were just like any otherteenybopper pop group with a good gimmick’ (McCormick 2003)."
So I think part of the answer for t.A.T.u.'s brief popularity is actually homophobia: especially in the 90s and early 2000s, for the most part lesbianism was kind of not really viewed by the mainstream as 'real' beyond a fetishized, sexy spectacle of 'hot young women playing with each other' for male entertainment, so it wasn't taken as a threat to the status quo. This isn't to say that their existence didn't also help and inspire a lot of Russian LGBT people, who chose to embrace t.A.T.u.'s queerness at face value and drew strength from that. As a bisexual teenager when they became popular, I also felt excitement at finally seeing two women kissing on tv. But ironically, though, I don't think a band like t.A.T.u. could exist in Russia today without being persecuted precisely because Russian LGBT activism is growing and now becoming a bigger threat to conservative values.