Does anyone know where the SQL Server Connection Properties are stored?
The list of the servers are kept in the msdb database.

SELECT * FROM msdb.dbo.sysmanagement_shared_registered_servers


If you're wanting the connections and tabs to be the same, then you would need to go to Tools --> Import and Export Settings... and export the selected environment settings. 
The two that would work for what you're asking for is under Options --> Environment. The two options would be "General" and "Tabs and Windows". It will export to a file. Send that to whomever will need it, and they can import it.