I've used a lot of exchanges that required a verification for creating an account with them, which is really getting me really annoyed and it did break the anonymous rule, which isn't good, besides every few years they require additional verification.
Many exchanges do not ask for verification unless you are going to withdraw a huge or the maximum amount or you can withdraw, there are levels on the amount you can withdraw and that's where the KYC comes in, you can use Decentralized Exchange like Uniswap if you really want to be anonymous and with no KYC, for me, it's not about the KYC but the reputation of the exchange that you are trading.