Any centralized exchange can ask you for KYC at any time and for any reason. Even ones like Binance which widely advertise their "no-KYC accounts" regularly lock accounts and freeze funds and demand KYC. Whenever you use a centralized exchange, you take the risk of either completing KYC or losing your coins.
The only way to ensure the safety of your coins and to ensure you will never have to complete KYC is to use a decentralized exchange such as Bisq.
For p2p trading, there will be additional verification, in a way you will be asked to snap your real face. Making use of credit card to purchase is another KYC as you are linked to your bank account.
It is entirely possible to trade peer to peer without giving away any personal details whatsoever. You just have to be considerate of the fiat payment method you are using and the other party you are choosing to trade with. Some will ask for ID verification for their own reasons. If you don't want to provide that, then just don't trade with these people.