Here, allow me to make this a bit more clear as to why she doesn't lose the bear
A. Her animation when ulting does not have her throw/drop the bear (clearly)
B. Her lore refers to a shadow bear, not a stuffed one. Personally, I believe that Annie had the stuffed bear before getting the shadow bear, she could have then used the stuffed one as a voodoo doll, I guess you could put it.
C. If Annie did happen to lose her stuffed bear when using her ult, that would mean that there would need to be a cast time to the ultimate, which would be quite the nerf to her.
EDIT : Not entirely knowledgeable on Annie's lore for B. I assume it is correct