I do acknowledge that generally it should be easier to understand what the question is about when the tag shows the full name of a product than when it shows an abbreviation.
However, in my view that argument's weight is lessened in this specific case by the fact that the SSMS abbreviation is fairly well-known in the SQL Server world. And we do show the expanded name in the tag guidance for those who do need an explanation.
At the same time, there are arguments against the rename. First, ssms is much, much shorter than sql-server-management-studio. Because of the dramatic difference in length, it could take noticeably longer to scan the tag section if we go with the proposed change (again, my personal opinion). Not the end of the world but the difference might still prove annoying.
Secondly, when editing the Tags section of a question, I like to take advantage of the list of suggestions that shows when I start typing the name of a tag. Now, there are many sql-server-nnn tags already, and the suggestion list shows only 6 items, which are chosen based on their popularity. Currently the number of SSMS questions doesn't make ssms one of the top 6 SQL Server-related tags. However, that might well change in the future, which would mean that sql-server-management-studio would start showing in the suggestion list along with tags related to SQL Server versions or concepts when I start typing
sql-server-. Now, fair enough, you will say, if it's popular enough, why not. My response to that is, a tag for something that can work well as an abbreviation should not force out a tag related to a thing that cannot be abbreviated.
And finally, something that isn't a counter-argument per se, but still. If we go with the proposed rename, shouldn't we be consistent and rename other tags, like e.g. t-sql, ssas, ssis? Expanding all abbreviations might make some questions clearer to less familiar audience. I expect that it will also reinforce my two counter-points above, though.