Huh, I have just never heard of these things, and I kinda feel like I should've. Personally I was more curious as to how the phrase became a thing.
The most likely answer would be that furries wanted an easier way to say "furry persona," kinda like how the Japanese use "persocom" for "personal computer." Even so, it seems unnecessary to me, since "persona" covers what "fursona" covers and more.