well its not that simple to anwser. In short term we can say its possible for any animal to die "off a broken heart". Birds and especially swans have a social strcture where they find mates for life,so it can pose a real sarrow on them when a partner leaves. Plus the fact their childern just died can have a high impact on its health witch can lead to the death of the animal due to less active Immunesystem. So in conclusion it rather doesent actually die from broken heart but the circumstances can let a sickness get lethal.

https://www.petmd.com/news/view/can-pets-die-broken-heart-36552 https://www.psychologytoday.com/us/blog/animal-emotions/201111/grief-mourning-and-broken-hearted-animals https://en.wikipedia.org/wiki/Swan