Lilo doesn't actually think that Stitch is a puppy - rather, she runs out of that dog pound screaming, then tells her parents to call the Men in Black. They arrive in Jetskis, and Stitch is vaporized into ashes. After all that happens, the family has their minds wiped by the memory pens, then the men disappear. Lilo then goes about her normal everyday life once again. If it wasn't obvious, this was actually a failed attempt at creating a Men in Black animated film.
Close Encounters of the Third Kind