Key

• the private key remains (only) with the user (The possession of this key is proof of the user's identity. Only a user in possession of a private key that corresponds to the public key located at the server will be able to authenticate successfully.
• the public keys are stored on the server in a file known as the SSH - Authorized Keys file (on Server)

For the procedure. see Digital signature procedure

Key Manager

The KeyManager is a program (or function) that decides which authentication credentials should be sent to the remote host for authentication during SSL handshake.

Mutual

• Mutual Authentication, in addition to server authentication, the client also has to present its certificate to the server.
• The server verifies it
• If both server and client authenticated themselves, then authentication is a success.

See the dedicated page: What is a certification base authentication (also known as Client certification / Mutual TLS authentication) ?