I do not think that this can be done in an SSMS object.
Some nodes in the view tree allow you to set a filter, but this does not seem to apply to the list of databases.
See if you can force the hoster to deny VIEW ANY DATABASE permission. You can also point out the problem and assume that they simply remove this permission from the public role.
BTW in Object Explorer, typing the initial letter of your database, at least moves you to the right side of the tree (provided that they are not all named with the same initial letter, of course!), And from a quick test of this end, entering the whole name really translates you to the correct database.
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