When you visit a server using mutual TLS, you'll get a prompt showing all your client certificates that match CAs that the server accepts. Once you select one, all your future requests will use that client certificate and be associated with that identity.
The client certificates can even be placed into smart card hardware devices (which can be USB or actual smart cards) that require a PIN or some other factor to use.
Because it's all built on public/private key encryption, the server has no credentials to lose in a data breach. Nobody can steal your credentials and reuse them to attack your other accounts.
And this is supported by all browsers today.