SSMS How to find the Connection name
The comment left by user sepupic is correct:


  And indeed there is no interest to server to know how did you name your connection or local registration, so only your client tool may know this


The only places I might have expected to see this would be the "connection" related DMVs, for instance if SQL Server kept track of the connection name used to connect to it (similar to how it keeps track of the client host name, set options used, etc).  

However, it's not present in any of the main ones I checked:


sys.dm_exec_connections
sys.dm_exec_requests
sys.dm_exec_sessions


And this makes sense, especially with registrations where the client has to translate those names before creating the connection anyway.