There are people capable of loving their pets as if they were their own children. Just because you're not, doesn't mean they can't. Losing things you're emotionally attached to is very painful. Just because you wouldn't have loved that particular "thing" doesn't mean they can't and doesn't invalidate their suffering. I know it's hard to see beyond your own pain, but that's exactly what empathy is all about. Maybe try it sometime?
> we just have a totally different set of moral axioms and are fundamentally incapable of seeing the world in the same way.
That set of moral axioms you hold was what allowed people - you spiritual ancestors, no doubt - to murder millions and millions of people of different skin color or believing in another god. It's what allowed people to abduct and enslave and murder whole populations for hundreds, thousands of years. The belief that "they are not people, they are beasts, so we can treat them like livestock" is what underlies nearly every atrocity ever committed by the "human beings" you hold so dear as a race.
Today, the set of humans seen as people expanded, which is a good thing. People seem to have learned that people everywhere are fundamentally the same: they feel joy, sadness, rage, love; they all bleed the same when hurt; they all live their lives enduring hardships and enjoying good things.
The thing is, all of that is also true for animals. They are capable of feeling joy and sadness, of getting depressed, of jealousy, of loyalty, of empathy, of fear and anger, of suffering. They cannot voice their opinions, but you know - there are disabled people who also cannot do this, due to disability or brain damage. Such humans are no worse as people than the others: their families love them, care for them, and grieve for them once they depart. Your "moral axioms" are then inconsistent: you should either promote all animals to human status or demote disabled people straight into gas chambers. Just so you know, we've already tried the latter, it wasn't great.
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None of the above should be particularly controversial, these are the basics, so let me finish by citing a poem of Andrzej Bursa I quite like, which (roughly translated[1]), goes like this:
Children are nicer than adults
animals are nicer than kids
you say that thinking this way
I'll find that the dearest to me
is paramecium caudatum[2]
so what
I prefer paramecium than you
you son of a bitch
[1]
https://wiersze.annet.pl/w,,11217[2] https://en.wikipedia.org/wiki/Paramecium_caudatum