Since I have several SSH keys in my .ssh/ directory, SSH tries to use each of it to connect to the SSH server. So when the SSH server has a very low "MaxAuthTries" setting configured, then the SSH connection may fail before password authentication is offered.
In order to connect a SSH servers with a low "MaxAuthTries" setting, you can use the following command:
ssh -o PubkeyAuthentication=no user@host.tld
After using the "PubkeyAuthentication=no" option, I could login to the host and add a SSH public key to the .ssh/authorized_keys file.