Probably after the hack, I guess you need to stay away from Kucoin exchange. Almost every crypto exchange asks our KYC for depositing fiat funds to prevent Money Laundering and trade them in crypto. There are a few ways you could trade anonymously like swapping your fiat to bitcoin in any Bitcoin ATM or using Localbitcoins and finding a trader to buy bitcoin using fiat.
But, from last year you need to verify your ID in localbtc if you are going to trade more than $1000 otherwise you need to give your mobile number for less than $1000 trades. Check their blog post for more info regarding this :
https://localbitcoins.com/blog/id-verification-update/ . Other than this, I don't think there is a way for you to buy bitcoin anonymously without handing over your ID.
There are possibly a lot more risk factors associated while handing over your ID proofs to a crypto exchange, you can have a read about that here :
Why KYC is extremely dangerous – and useless and here are a list of P2P exchanges where you can trade anonymously :
List of P2P/no-KYC exchanges. You should stay away from selling your KYC for just converting fiat to bitcoin, rather use P2P exchanges like Bisq :
https://bisq.network/