We need a bitcoin hardware wallet kind of password manager, where the actual passwords are stored on a hardware security key. When you click on the computer on the password you want to use, the hardware security key shows it's name on it's screen, and asks you to press a button on it to confirm that you want to use it.

For backup, the hardware security key let's you download a file from it with all of your passwords encrypted, and the decryption password it's shown on it's screen (something like 12 random words)