Certainly not the classic Walt Disney (and current family) version where his wife was the person who changed Mortimer to Mickey since she didn't like name Mortimer and the name may or may not have had been related to a Toy already named Micky and the fact that Disney like alliterative names
I'm not sure there's any way to confirm for sure if this Mickey Rooney version has any grain of truth, but knowing Rooney's history I'm guessing it's a fanciful story he told and embellished for many decades.
I do think Mickey Rooney's popularity in early Hollywood films and shorts very likely had a bit of effect on the popularity of both MM and MR
The Mickey McGuire shorts had been pretty popular for 2-3 years prior to the first Mickey Mouse so I'd say it's possible, but
still seems to be unlikely. Rooney's dates have varied from mid 20's to just 1920's, but that certainly doesn't change the chances, but
probably deserves to be a fun story and urban legend rather than a true telling of the events.