That makes your data connectable to your blockchain address. In that case it’s also still traceable if you send your funds further to multiple new addresses - they’re all connected to the withdrawal.
To achieve total privacy I believe you would have to buy btc in person for cash and have it sent to a newly generated address. In that case it’s still a risk that the seller’s address is connected to his personal information and in case of a thorough investigation he can be traced and disclose information about you.
Theoretically another option could be also to pay someone (for example a homeless person) to complete the KYC on the exchange in their name and let you use the account.