How to update SQL addins to work in SSMS 2016?

Indeed, .Addin are no longer supported. You need a VSIX project building into "C:\Program Files (x86)\Microsoft SQL Server\130\Tools\Binn\ManagementStudio\Extensions[Your extension name]". There is not much information on how to extend SSMS 2016 but:

Here is some documentation for VSIX projects: https://docs.microsoft.com/fr-fr/visualstudio/extensibility/index

That forum helped me a lot: https://www.sqlservercentral.com/Forums/1802009/Developing-Extensions-for-SSMS-2016

You can't use Connect.cs anymore. The link between SSMS and your code now works with commands and packages as explained in the second link I posted.

Hope that helps