Decentralized exchanges can be anon but you can also have centralized exchanges that is as anon as it gets without sharing your KYC or even your first name, just a mail would be enough for many of them.
Exchanges that don't subject you to KYC verification aren't anonymous. They can still link the information that they do have to certain events that you are connected to. On top of that, they can subject you to KYC verification out of nothing, and we have seen that this is happening more frequently than people are comfortable with.
Your information will be stored on random servers for ever, and when these "anon" exchanges you're referring to go down, they'll very likely sell your information to make a last quick buck.