If they're not the same person, Pixar's getting pretty lazy at character design.
True, Jessie and Andy's mom don't recognize each other, but it's been a long time. There's no reason Andy's mom would realize Jessie was her exact same childhood toy, and there's no guarantee Jessie would recognize Emily as an adult. She's used to living life as an immortal, unchanging plastic being, so the idea that her old friend could morph into an even bigger giant with increasingly decaying flesh on her face must seem pretty bizarre. Finally, as you may have guessed from the fact that we keep calling her "Andy's mom," the character's name is never mentioned in the movies, so it could easily be Emily.
Sure, the creators themselves have seemingly shot this theory down, but this is Pixar we're talking about -- they could simply be lying to try to maximize the amount of emotional damage they'll do to us with the next movie.