A lot of people refer to Call of Duty Black Ops 3 as "COD BO3" to shorten it to 7 characters.
I think this is worth doing. There's certainly no harm in adding a synonym - it's not going to change the current use of the tag. In fact, synonyms are meant for common, alternate spellings or phrasings for tags, so that people can find them easier when writing questions or searching tags.
In fact, the Call of Duty series has a long history of being shortened in the way you're suggesting by the CoD community, and Google recognises them and gives similar results on searches.
I don't see the need for this at all. You can refer to the game with the shortened terminology in the question, but a shorter tag doesn't help searchability at all.