I have created a new login with read access to 2 databases.
The user reported he could login to the instance suing SSMS but not see either of the databases in Object Explorer.
He could only see system databases and database snapshots.
I checked his permissions and they looked ok.
The login was integrated authentication. Just as a trial I created another user for him as a SQL Server Login but he had the same issue.
If he launches a new query window he can Use Database and then run SQL queries against both the databases I granted access to.
So it looks like the problem is confined to SSMS.
I don't know if this is a bug or if there is some configuration setting I am missing.
Picture below of what can be seen in object explorer.