I collected some information into this.

Basically, this occurs when I am closing SSMS. Sometimes, right after the interface closes, it restarts immediately in the second that follow. So, it all turned out that it is a keystroke effect into RDP. Basically, the last icon used is SSMS. So, it has the focus. When SSMS closes, for some kind of reason, when I close RDP, an Enter key is sent. Then, when I connect back later, SSMS was restarted and waited since then at the Connect dialog. As I mentioned, this only happens a few times. It is very rare.

There is something wrong causing this Enter key to be sent. This is the only environment I am using where I see this behavior. It happens occasionnally so this is just not a bad typing I would have done. I am also closing with the mouse so there are no keyboard keys being typed.

Then, Event Viewer kicks in and logs two errors.