Handkerchiefs are associated with sorrow and pain, which I see a lot of in the Animal Doctor's world. Some go into fits of bawling and sobbing ( this is me) while others can handle their grief silently like this old woman who listened intently, tears rolling down her cheeks, as the Animal Doctor explained how he and the other vets had tried to revive her dog. Her husband was beside her, rubbing her shoulders while his other hand clutched at a rosary.
I also remember this guy whose Labrador suffered from heatstroke on the day of his wedding. Before leaving for his honeymoon abroad, he went to the clinic, sat by his dog's crib and murmured to it for several minutes. He tied a handkerchief around its paw, hoping that the dog would still be able to smell his scent on it and come out of coma. But the dog never made it.
I remember these images very well, perhaps because no words can accurately describe the bond between a human and his pet, and unless another person has tried it, no one would probably understand how loving an animal can be both so incredibly beautiful and painfully sad.