u/noproveryay has some good suggestions.

One thing I'd note that the 1990s (or for our purposes here the 1991-2001 period) is still mostly covered by social sciences like political science, and by journalists, rather than academic historians proper. Frankly, academic history of the USSR is just beginning to move into the 1960s, but that's another story.

There are a few exceptions I'd add. Stephen Kotkin is a well-known Soviet historian, and his Armageddon Averted covers the period 1970 to 2000, although it's more focused on the end of the Soviet period than the 1990s. Serhii Plokhy is another historian who has written a number of books that touch on the late Soviet and post Soviet period, although he is more focused on Ukraine. His Last Empire is a good microhistory of the events of 1991.

As far as journalists go: David Hoffman's The Oligarchs is a good insight into just what went down with privatization in the 1990s, although the book itself is about 20 years old now. Steven Lee Myers' The New Tsar: The Rise and Reign of Vladimir Putin is a biography of the Russian President, but gets into some useful background of what happened in the country in the 1990s and how that shaped the economy and politics that came after (this insight Hoffman obviously wasn't able to do at the time).

There's a bunch of political scientists (among others) who have written some different introductory texts for students on Russia since 1991 which might be of interest. Works by (or edited by) Stephen White, Archie Brown or Richard Sakwa come to mind.

I think modern Russia steps too far into the twenty year rule for the forum, but several recommendations for the 1990s come to mind.

Svetlana Alexievich, a winner of the Nobel Prize, addresses this very question of what it meant to be Russian in the 1990s in Secondhand Time. Yegor Gaidar's Collapse of an Empire is an intriguing and provocative argument about the structural problems of the later Soviet system (Gaidar, as Prime Minister, oversaw shock therapy, so his perspective is particularly interesting but should be read in that light). In that spirit, Yeltsin's Midnight Diaries is invaluable to explore just what led to the rise of Putin at the close of the millennium.

Another provocative, insightful and remarkably close analysis of one of the major scars of the 90s, Chechnya, is Anna Politkovskaya's Small Corner of Hell. Politkovskaya was assassinated in 2006 (without going too much into conjecture, Ramzan Kadyrov and Vladimir Putin are central figures in her case).

Let me know if you had something else in mind for post-Soviet Russia and I am happy to recommend additional texts.