Not really. If you want to go back a Decade, then it was legitimate equality issues.
You can't just say, people shouldn't be equal and claim you are fighting against 'identity' politics. Like women being allowed to open bank accounts without their husbands permission. Why do Republicans want to go back to those days? Unless you actually listen to them, and they quote some Bible Versus about Women being property, then you see the actual agenda.
What is the term for 'take a couple isolated examples and call it a movement'? motte-and-bailey goes both ways.
There is no widespread 'trans women competing against biological women'.
Completely made up issue to stir outrage with the radical base.
It's more common than you may think - see https://www.shewon.org
There are actual real genetic disorders Trans people are dealing with.
Republican's just chose Trans people as some small group dealing with a difficult to explain condition and decided to pile on them.