Been a while since I watched Cowboy Bebop, but I think it was simply meant to express how tenuous the bonds between the Bebop crew were. Jet was always the sullen proprietor who never was too hyped about having anyone other than Spike, Faye was an outsider through and through. They accepted Ed's presence because she brought some life and she was definitely useful, but she was always some kid who would be better served being with her folks or what served as such.
Spike always tried to be the cool guy when all was said and done, so he gave her some toy because it seemed like the cool thing to do and he had no particular grievance about Ed being on board.
My takeaway from the series was always that nothing lasts forever, people come and go, shit happens with no particular rhyme or reason. As an audience we want to grow attached to the characters, but the show was about how, were such eclectic characters be real, they would naturally grow apart or, well, die.
With that said, here is best girl.