All right, I am at my wits end.
I have followed the link in the "Connection failed" picture and it brought me here:
I have read the whole thing, I was able to run portqry.exe and it did return information and contained my target instance, so I can rule out points 4 and 5.
That leaves me with first 3.
My hostname and IP, as written is: "USER\SQLEXPRESS" where "USER" is my pc username all in capital (I am trying to connect to my localhost db), so I can rule out point 1. and I have tried out point 2. ("USER\SQLEXPRESS" or "USERSQLEXPRESS") to no avail.
Point 3 - I was able to ping myself in CLI with command "ping user" so I can rule out point 3. as well and thus have no idea, what causes that timeout.
Does anybody have any idea what might be causing this?