"Furbaby" arose to denote pets considered as family by devoted pet owners. Some devoted pet owners say they love their pets to the point that some people love their kids.
The term "furbaby" has drawn much ire from human parents of human babies. There are internet article titles such as, "Saying Your Dog Is Your 'Baby' Is an Insult to Moms Everywhere," "Dear Pet Moms, You Are Not A Mom," and "Why I Hate The Term 'Furbaby'". They tend to argue that one feels a greater connection to a human child, that looking after a human baby involves a heck of a lot more effort than looking after pets, and that calling a pet a furbaby is trivialising the amount of effort and love a human child requires to the point of being insulting.
People on the furbaby side often point to the fact that their pets mean a great deal to them, that the pet is wholly dependent on them, and some say they would even be willing to die for their pet. Some furparents are unable to have human babies of their own and their pets therefore become a kind of surrogate.
Overall, my opinion is that the term is harmless. No one is literally suggesting that a pet is identical to a human baby, or that human babies don't take a vast magnitude of more effort. I am not a rabid pet parent, leaving most of the caring to my partner and often ignoring our pets, but even I will occasionally tell a pet that Mum & Dad are home. The word "baby" is already used casually, such as when adults use "baby" as a sincere term of endearment to their romantic partners. Saying "furbaby" is not such a great leap for casual use.