My personal opinion (which is I think at least loosely supported by research into animal vs. human memory) is that relative to humans, animals are very grounded in the present so they grieve much differently.
When a human's loved one dies, we miss them in the present moment... but we are also thinking of the past and future. We think of our days ahead without them and mourn the happy times in the past that can now not be repeated.
It was surprisingly hard to find a scientific
consensus on this topic.
There's no real objective way to measure feelings, particularly in animals.
Even in humans, you usually just have to ask the subject how they're feeling.
Behavior changes are at best kind of a decent proxy for making... educated guesses instead of uneducated guesses into an animal's feelings.
We could measure things like oxytocin and cortisol levels or take fMRIs, but good luck collecting those metrics without profoundly altering the animal's emotional state.