The problem with all debates of "identity politics" is that there is no clear definition of it, not even by Mark Lilla who popularized the term in 2016. (Lilla, by the way, doesn't even speak of or for the "left", but of two types of "liberalism", one that he supports and one that he rejects.) "Identity politics" is a textbook strawman argument which any decent analytic philosopher should be able to tear into pieces with propositional logic. What's more, the term has become a reactionary meme now that political movements, such as "Aufstehen" in Germany, are being founded on the premise of reinvigorating the left by ridding it from "identity politics". This is where the strawman becomes a red herring.
All this is mostly based on the fiction that the working class defected to the extreme right after established left-wing politics no longer represented it. It's a fiction because, at least in Europe, research has clearly shown that most voters for the extreme right come from the middle class and vote for these parties because of shared core values (in short, an understanding of the rule of law as law and order, and an understanding of democracy as the execution of the will of the people who represent the majority population), not policies.
If Lilla and others were more consequential, they would have to historically denounce the political left as "identity politics" as such. One could call the French Revolution "identity politics" of the bourgeois (versus the aristocracy), the 19th century workers' movement "identity politics" of the working class (which an old-school Jacobin might have rejected precisely on the grounds that the republic had declared everyone to be equal), the feminist movement "identity politics" of women, the black civil rights movement "identity politics" of African Americans, the gay pride movement "identity politics" of queers etc.etc.. In the end, those who deplore "identity politics" express a nostalgia for a simple, binary past that never existed. Worse, they patronize groups of people to which they neither belong, nor are in touch with.
Maybe there could be a more precise notion of "identity politics" in the sense of political choices purely made on the basis of one's group identity instead of one's political interests. Examples could include trade union members who voted for Clinton, Blair and Schröder in the 1990s out of token loyality to "their" party, or the blind support of openly destructive and malicious politics on the basis of ethnic loyality in areas with ethnic conflicts. In my hometown Rotterdam, for example, a right-wing populist party has been the strongest political force for one and a half decade simply on the basis of white ethnic voter loyalty (in a city whose majority population is now non-white), never mind the fact that this party is chasing its own voters out of the city by aggressively gentrifying traditional neighborhoods. Did Lilla and his epigones ever call this "identity politics"?
-F