Q: James’ Patronus is a stag and Lily’s is a doe. Is that a coincidence?
JK Rowling: No, the Patronus often mutates to take the image of the love of one’s life (because they so often become the ‘happy thought’ that generates a Patronus).
So why doesn’t Severus’s patronus become a stag instead of a doe….?
Lily loved James and was loved back in return, so her Patronus was equal to his—a partner.
Severus loved her but it wasn’t returned, and he more or less devoted his life after her death to protecting what was left of her. So his Patronus doesn’t draw on himself, because in that regard there’s nothing left of him. There’s only her.
Just gonna go right ahead and steal this headcanon…