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SOLVED! The machine is a Dell Poweredge system which ran what they call iDRAC in the background - iDRAC in its turn ran a SSH-server which conflicted with the one I installed in Debian. The solution for me was just to disable iDRAC since I don't use it, and I'm glad I don't. I have no idea how that conflict should've been solved.

15:08:26.493 Started a new SSH2 session. 15:08:26.493 Connecting to SSH2 server 174.xxx.xxx.xxx:22. 15:08:26.493 Connection established. 15:08:26.681 Server version: SSH-2.0-OpenSSH_5.8 <---- 15:08:26.681 First key exchange started. 15:08:27.289 Received a new host key from the server. Algorithm: xxxxxxxxxxx 15:08:27.320 First key exchange completed using ecdh-sha2/nistp521. xxxxxxxxxxx 15:08:27.320 Attempting password authentication. 15:08:28.724 Authentication failed. Remaining authentication methods: 'password'. 15:08:30.581 Authentication aborted on user's request. 15:08:30.596 The SSH2 session has been terminated. 

Status for networking.service gives shows this

15:08:26.493 Started a new SSH2 session. 15:08:26.493 Connecting to SSH2 server 171.xxx.xxx.xxx:22. 15:08:26.493 Connection established. 15:08:26.681 Server version: SSH-2.0-OpenSSH_5.8 <---- 15:08:26.681 First key exchange started. 15:08:27.289 Received a new host key from the server. Algorithm: xxxxxxxxxxx 15:08:27.320 First key exchange completed using ecdh-sha2/nistp521. xxxxxxxxxxx 15:08:27.320 Attempting password authentication. 15:08:28.724 Authentication failed. Remaining authentication methods: 'password'. 15:08:30.581 Authentication aborted on user's request. 15:08:30.596 The SSH2 session has been terminated. 

Status for networking.service shows this