I have an old machine that has been correctly connected to a server that has a database on it. I have a new computer that I am trying to connect to the same server and database. The old laptop has under “Database Engine” a list of servers (ex. “COMNAMEMSSQLSERVER”). Here is a screenshot of SSMS on my new laptop. I am not understanding how you connect a server to SSMS and have it show up on this list.
That list, as the label says, is Local Servers.
If you are trying to connect to a server elsewhere, you need to enter the server name in the previous dialogue. Unless SSMS is on the same machine as the SQL Server, it will never show up in the Local Servers list.