Does Lily have any special role, or is she like just a pet?
Pfft-
Funny as it is to imagine Lily as a rabid hound that Team Haven occasionally releases, no. Lily has more of an active role as well. Believe me though, you're not the first to think she's just a little feral stray that Scott adopted, @solarspinel literally drew Lily as Scott's backpack leash kid at one point.
Lily's role in Team Haven is made to further expose how conditioning and other stuff still haunt the other toys, especially the Bigger Bodies there. She's the answer to the question: How can I make sure that the Fledglings' suffering as the survivors don't completely overshadow the suffering of the orphans who actually went through the initiative?
Lily's presence forces the toys to confront their trauma all over again in the present, and for the Fledglings to see exactly what that looks like instead of just recordings and documents. This time, it's not this past event that they're witnessing through VHS tapes and recordings. This time, it's a plague of trauma that's actively challenging the foundation of Team Haven.
And it challenges the Fledglings too in a way. Because how do you comfort someone who went through something when you're 'the lucky ones who got away'?
Lily is perfect for bringing the true tragedies of Team Haven's toys forward, simply because well... She's Lily.