There is no point in hold that Bitcoins if it is publicly known who is behind the address, so best move is to return that coins to exchange and maybe they will reward you because you honestly returned something that was stolen.

Personally, I wouldn't mind keeping some coins that some rich user is send by mistake to my address, but hacked coins from exchange is something else entirely, because they belong to many small users.

Not using KYC is very difficult nowadays, but changing BTC address is very easy to do. But on this forum we do not have KYC, so signed message does not constitute a breach of our privacy. But we are not completely anonymous on this forum if we not use TOR or VPN, and in case we do something bad and using our real IP, admins can give such data if they receive a request from the relevant services.