I personally even think that kyc data is much more safe on gambling casinos than they are on crypto trading exchanges, this is because from what we have seen so far, I've come to believe that crypto trading exchanges are more prone to cyber attacks than gambling casinos, I've rarely come across any news of a gambling casino being hacked and users kyc data stolen, but this is something that is particularly very common with crypto trading exchanges.
And concerning how kyc data are kept, I think some platform does have a special server, more like a storage system where every kyc data is uploaded to and kept, and most times, it's only the CEO, and maybe some else very close to the CEO, that usually have access to such server or storage system..
I agree with you and for avoidance of doubt or misunderstanding take the FTX case for example, the CEO said that the collapse of the exchange os as a result of manipulation of its system, even though it is clear they spend the exchange liquidity on personal expenses, but it was aided by the hack which can lead to data leakage from The exchange KYC database.
And there are some other cases of exchange hacks and all these hacks will expose user data to identity thefts.