So let's assume an address has been made public and everybody knows who that address belongs to, can the owner decide to change that address and move his coins to another address without it showing on the bitcoin blockchain?
Every transaction will always be present on the blockchain. The only way to hide the transaction is if you used a mixer service so your transaction will be mixed in with other transactions to only show as one transaction on the blockchain.
I mean can the wallet simply change the address at the request of the owner with verification like the seed phase, of course? Then on the blockchain the new address will not show where the bitcoin came from and the old wallet will just show $0 balance without showing where the coins where sent to?
The wallet can’t just change the address that’s already on the blockchain but the wallet can generate a new address.