I think that relationships have very fuzzy edges-- actually, a lot of overlap. The relationship with a pet and a child isn’t the same, but there’s some overlap, in that you have chosen to take on a responsibility, and are obligated to care for this creature. They also bestow boundless affection on us, and enthusiastically greet us, like children do up until about the age of four.
There’s other overlap to in relationships that aren’t in any way the same-- and if you said they were the same, it might even be a little creepy. For example, though, most people are in the same age cohort as their spouses, and that is something they also generally share with their siblings. So there is a lot of cultural “touchstoning” with both spouses and siblings. Then there is the way you feel about your siblings and your parents, which isn’t the same, but there is overlap, in that you share a deep history.
There’s spouse/child overlap too, in that these are the people you live and die for.
So I think it’s pointless arguing whether pets are like kids or not like kids. It’s not one or the other; all or none. My son is more important to me than any of my pets, but to be honest, my pets are more important to me than any other child in the world other than my own son.
I don’t like the term “furbaby” either, because there’s something cloying about it, but I get the concept. I just lost a cat about six weeks ago, and it was devastating. My own pets’ deaths have always been huge blows. Losing my parents hurt, and a few other close relatives, but generally, the loss of humans does not upset me as much as the loss of one of my own pets. People who aren’t pet people don’t get that. When an acquaintance, or coworker, or something, dies, it’s not as upsetting as losing one of my own pets.
I don’t think of myself as my pets’ parent, but I do take my responsibility toward them very seriously. I am no more lax in keeping up their vaccines than I am in keeping up my son’s. I am diligent in keeping their dishes and areas clean. Once when the cats’ food bag got wet, I made a trip to the store in the middle of the night, so they wouldn’t have an empty bowl-- but of course, the stores that were open didn’t have their premium brand, so I bought a small bag of Iams, and got their regular stuff ASAP. I know a lot of people who think “Oh, the cats can go without for a half a day.” But they* shouldn’t.* It’s my job to see that they don’t.
I don’t know if I’m making my point clear. Ask me questions if I’m not.