If you have the private key of your wallet then you can use that private key to sign a message. Maybe you have seen the PGP keys out here in this forum before, those are encrypted messages which can be viewed with your public key but can only be unlocked or in this case signed from your private key. And as we know private keys are what truly makes you the owner of the Bitcoin address.
https://bitcoinmagazine.com/articles/bitcoin-address-sign-1399914228/Try this article to find out more about how this signing addresses works and how to perform them.
To use a signed message as a proof of ownership of a bitcoin address, you need to disclose the public key behind that address otherwise how the other party would be able to verify your signature?
Disclosing public keys is not recommended practice in bitcoin community (don't take bob123 much serious

) it is why we discourage address re-use, Actually a very effective proposal about signing multiple utxos (with a same output address) using a single signature has been abandoned just because og its potential of encouraging address re-use.
PGP keys typically use very higher security levels (like 4096 bits) compared to bitcoin ECDSA 256k1 and it is why people are more relaxed about sharing their public keys.