The census form[1] does ask about ethnicity, sure (item 14; while you’re free to leave the field blank or write “Martian” there if you wish, it’s true that most people won’t). It doesn’t ask for your
name, however (and isn’t received or sent by post, either—census takers are supposed to go door to door). Not that it’d be difficult to recover if you intercepted the forms before they were collated (despite assurances of confidentiality printed on them), I’ve just never heard of anyone trying that or even being vaguely interested in it.
The story you’re telling sounds like something out of Kanevsky&Senderov[2]. I’m not denying it might be happening, but I’ve never heard of it and would be interested to hear more—at least a location and a year would be helpful. I’ve never filled in, seen, or heard of such a form in relation to higher education, but then I only have personal experience with Moscow and second-hand one with St Petersburg and Ekaterinburg, all large cities without much of a sharp divide between common nationalities of residents (as in Bashkortostan or Chechnya or a number of other places).
(To call Moscow ethnically homogeneous would be a huge stretch, mind you. I can name [former?] Moscow residents in my contact list with Armenian, Azerbaijani, Chechen, Jewish, Korean, Mordovian, Roma, and Ukrainian ancestry without even opening it—hell, I could nominate myself for a third of those slots. I can’t even imagine what I’d learn if I actually went around asking my acquaintances about their family history. I suppose most of the results would still count as “white” by US reckoning, but, well, meh to that.
Culturally, though, yeah, things are pretty uniform. I just wanted to warn you away from thinking that that 80% figure reflects the country’s people mostly coming from a single ancestral group as opposed to them being subjected to pressure to have “Russian” written in their ID for like half a century. There’s a reason why optimistic discussions of Russia’s medium-term future usually include the question of the number of states that would exist in its place.)
[1] See link to PDF file at http://government.ru/docs/38324/ for the version used in the botched 2021 census (labelled 2020, this being a document from back before Covid).
[2] https://www.worldscientific.com/doi/10.1142/9789812701169_00..., free at https://www-users.cse.umn.edu/~shifman/EinsteinBook.pdf