It's because pets are often a major source, sometimes your only source, of comfort through difficulty.
No one has ever explained it to me this way before, but now that I am experiencing it, it's very obvious. I'm incredibly sad and normally when I'm very sad, I sit with my pet (or play with them, or take a nap with them) and their companionships and familiarity is a great source of comfort. So the pain is just compounded.
I have lost loved ones before but never someone I was currently living wither who had been a great comfort to me. It's a very specific thing. All I want right now is to give my furry friend a hug or see their sweet face or feel the weight of their warm body next to me. And it's the one thing I can't have.
I feel bereft.