When you import an existing wallet using your recovery phrase into Coinbase Wallet extension, there is no impact on your existing wallet. You will be able to continue using your existing wallet across multiple providers, and any transactions you make will show up in your wallet across all of your wallet providers.
While the recovery phrase is portable, the password that you set for your wallet is provider-specific and will not be ported over to another provider. Like recovery phrases, passwords cannot be recovered by any wallet provider, including Coinbase Wallet.