I assume OP is talking about sites where you create an account, deposit to that account, then start betting (Cloudbet, Anonibet, Bitcasino, etc) not dice sites and sites like Directbet and Nitrogen where you don't need to make an account - generally on these sites (Cloudbet etc) you will be able to start betting as soon as you've made your deposit, with no confirmations required, but won't let you withdraw without three confirmations.

The reason is that when you make a deposit there's no double spend risk to the site - you can start playing straight away, and even if you have double-spent your coins they don't lose anything (the worst that can happen for them is that you lose your entire balance to them, but then it turns out your balance wasn't legitimate to begin with so they don't actually make any BTC from you). However, if they allow you to withdraw some of your balance, all of your balance, or your balance and your winnings, and then they find out that you double-spent your coins, they'll have paid you out without ever receiving anything from you in the first place. By waiting for confirmations before allowing you to withdraw, they're ensuring that your deposit was legit.

TLDR: There's no risk to a site in accepting a deposit and letting you play straight away, but if they allow you to withdraw before your deposit has been confirmed they expose themselves to people double-spending.

EDIT: I think I misunderstood.. is OP asking about non-bitcoin sites? If so, then the answer is pretty much the same. If you bet and lose all your money then there's not risk to the site, but if you try to withdraw then there's a risk that you're actually laundering money - that's when the KYC stuff comes into play (ID, bank statement, utility bill, etc), it's part of their licensing requirements in order to stop money laundering.