Oh that's a good thought. It never occurred to me that Mimi could be pronounced my-my, but I'm coming from a place where I've seen it often enough in Japanese media to know the actual pronunciation of mee-mee. Mimmy sounds slightly different, but its closer to mee-mee than my-my.
That is, unless Mimi is pronounced slightly different and is from an entirely different source than the Japanese name. There's an unrelated English/Italian name with the same pronunciation and spelling, but it's entirely possible a similar name exists from another culture