Solved! Go to Solution.
I ran into a similar issue on my end and wanted to add some information in the event that someone has my experience. In my case, I had ArcMap opened when I installed SQL Express workgroup as part of the 'Database Server Workgroup' installation (this was for ArcGIS 10.4.1; however, I suspect that this feedback applies to probably most versions). After the installation, I tried to connect ArcMap to SQL Express using the database server, but received the same error message being reported in this thread.
I navigated to the TCP Dynamic Port and there was no special port number assigned (the default 1433 was being used as reported by documentation I found online for figuring out how to find the TCP information).
After a bit of time, I tried closing ArcMap and re-opening it (thinking that ArcMap was not seeing the new service on my Windows machine). After I did this and tried to connect to my SQL Express instance, it worked without issue.
I wanted to add this as another item in case someone falls into this small trap and spends a lot of time troubleshooting (perhaps Esri can write a check on the installation script to make sure no ArcGIS for Desktop software is up and running during the installation?).