They are leftists. The woke ideology has a clear geneaology to Marx. Of course, they are not economic class focused leftists by any stretch of the imagination - bourgeoisie champagne socialists who've re-engineered the Marxian thinking style to focus on social identities, which makes it really easy to purport to care for the oppressed. It's very high quality champagne socialism.

But it's still leftist in origin and in ideology. Saying the modern wokified U.S. Left isn't leftist is just a No True Scotsman argument. Leftism and rightism are not narrow, specific things, they're more like inchoate moral intuitions that are given more concrete form by ideology and realpolitik. Individual leftist or rightist ideologies are often contradictory with each other. But the underlying sentiment is usually still recognizable as one or the other.