I just experienced a difficult situation and I did not know how to react. My SO and me have a pet hamster that showed some worrying behavior and might have to be put down. She is very attached to him and she started worrying and crying intensely. While I like the hamster, I am not really emotionally attached. What made the situation worse is that she always projects the future negative while I project it positive. So for her he was already dead and for me he just had a small thing that a vet would be able to fix in no time.
While I don't know the future, but when he eventually dies I can see questions coming up like why are you not sad? Did he mean nothing to you? How can I prepare for this event and also show her that I care about her and her wellbeing.
I saw this question which is great and gave me some further reading. But most advice is written for outsiders and not really shared pets. Also the main point is listening to the person. While I did that in the above mentioned situation for most she was just crying and not saying anything, which makes listening hard. I also did not really know what to say, I thought of the obvious also mentioned in an article in the question of what not to do: "Like you could get a new one", but did not say it.
The main problem is that I have the feelings of an outsider but the role of an insider(since the pet is shared) which makes anything I say like "I am so sad about this" fake or "My condolesence" too distant.
What I tried so far is just holding her and not saying much. I just said "That it will be fine" but this definitely won't help when he actually dies. Is there something more I could do than just physical contact, that could help her with her emotions?