Im not sure about that, since in the legend it was about a lonely wolf asked an injured deer to accompany it in exchange of sparing its life
I think its just that any white deer must married and mated into the royal family according to the custom and here the available one happened to be toguto, or something
Based on all the stories, the white deer are rare and those that are found are mated/engaged to the royal family member closest to their age. Most of the stories we see are of the male white deer, but there are female white deer and since white deer are generally rare, most princess and princesses marry other wolves, continuing the bloodline. In this story there is a female white deer but in other stories, other princes are shown to be with female wolves.
I thought all white deers could get preggyy regardless of their gender, but I guess I got it wrong.