I assume they take it as disrespect that their pet is an animal that is different than them. Rather than accepting they are simply another form of life with whole different motivations and concepts.
All I'm saying is that I don't think the word "grief" is the correct word in this context, because grief implies an understanding of the concept of death, which is a very human idea that we had to learn. We weren't born with this understanding, learning about death is something very upsetting that children have to go through.
You may believe it is not so, but maybe you now understand my position?
-- https://www.apa.org/topics/grief
That seems to describe the behavior people have seen cats display after the loss of someone or another animal in their homes. Maybe they don't know it's death, but they know it's a loss and exhibit signs of distress, anxiety, and confusion. We can't really ask them to find out if they're dwelling on the past or apprehensive about the future though.