Being anonymous doesn't mean you avoid revealing personal data such as your name, phone number, passport etc. Anonymity is the condition where there's NO WAY for an individual to work out your personal information. For example, if you give someone your IP address, you are not anonymous. The fact that your name remains unknown to them doesn't mean it will remain forever. There are ways one can locate you based on your IP address; first ones? The government.

The wallets you mentioned are not recommended if you want to remain anonymous. Instead, the greatest anonymity you can achieve in Bitcoin is by using a non-custodial wallet which is connected to your full node which is also connected to the Bitcoin network through Tor.

That's the highest decree of anonymity Bitcoin offers you.