How do you install a TFS plugin for SQL Server 2016 Management Studio?
Source control integration is being deprecated in Management Studio according to Deprecated Database Engine Features in SQL Server 2016, but that doesn't explain why it's not working in 2016. If you want it, I suggest you vote for this Connect item.

This can now be enabled by modifying the packages SSMS 2016 (Visual Studio 2015 shell) will load. See Source Control in SQL Server Management Studio (SSMS) by Ken Van Hyning (Engineering Manager, SQL Server Client Tools).

You will need to install Visual Studio 2015 (the free Community edition will do) in order to have the packages installed on your computer. You then modify C:\Program Files (x86)\Microsoft SQL Server\130\Tools\Binn\ManagementStudio\ssms.pkgundef to remove (or comment out) the lines relating to TFS SCC Configuration.

Note that on every SSMS update (which as you probably know are pretty frequent), you'll need to re-apply those changes.

I have put this process in a PowerShell script file. Trying to get it published to the PowerShell Gallery, but in the meantime you can get it from GitHub.

The same issue exists with SSMS 17, with the same solution ($SsmsUndefPath will need to be updated from 130 to 140 in the path).

Update Azure Data Studio offers an extension for source control. What is Azure Data Studio