Obviously, from my username, I guess I do use the "mom" thing! However, I'm not nearly as devoted to the terminology as some people are. I use it because "master" seems way too domineering, and "owner" too impersonal, although I do use the term "owner" occasionally. If I could think of a better alternative, I probably woudn't use mom. And I NEVER say furbaby or furkids, for what it's worth!
I rarely use "mom" w/r/t Daisy in public, and if I do, I get a little embarrassed. I definitely refer to her as my baby, though. And really, she's such a snugglebug that she reminds me of my babies at about 5 months of age. All soft and sweet and cuddly, looking up adoringly into my eyes, completely trusting, etc. OK, I'll stop now.
But believe me, I am a mom, a real human mom to human kids, so I do recognize the difference in roles between being a dog-mom and a kid-mom. (Although my kids accuse me of putting Daisy above them on occasion -- crazy-talk!)
My 10 year old daughter has talked about being an equine vet -- so we may have one very loving horse vet coming down the pike in about 20 years!