I am usually quite a practical person and don't consider myself very sensitive or emotional in that sense. Thing is, it's one of those things that unless you've been a pet owner for a long time its very difficult to comprehend and understand the loss and grieving associated with it.
My mum recently had her dog put to sleep. She was 14 years old and she'd had her since I was living at home. I can honestly say that the thing I missed most about living at home wasn't my mum or my sister, it was that dog. I grieved deeply for a period of several days. Should you be given time off work? Perhaps not. But I personally took a day off as A/L to help get my head around it and to be there for my mum who took the loss extremely hard.
To a dog or cat lover, a dog is not a dog and a cat is not a cat. They are companions. Yes, they are 'lesser species' and don't have any true emotional intelligence, but that is hardly the point. The point is they're not things, they are living creatures that can and do reciprocate affection, trust and companionship.