"Furkid" is a relatively recent nom d'affection used in conjunction with, or even in place of, the term "pet", when talking about a dog, cat, or other furry companion. Specifically, it is used when talking about a pet whose owner treats said creature like his or her own actual child, and the term itself is, frighteningly enough, also often used by these same owners.
This said, it comes as no surprise that the term "furkid" is often accompanied by extreme personification of the pet itself, generally in a familial, "parent-child" sense. Be it a discussion of grooming, hygiene, or disposition, the animal is referenced as if it were a family member, indeed, as the offspring or otherwise surrogate child of the owner(s). Furthermore (no pun intended), more complex, core familial issues, such as sibling rivalry are not outside the realms of this definition or behaviour.
The term itself can be compared to its lesser-used, but apparent (and perhaps disturbing in this light) contrasting equivalent, "skinkid". This latter term is, of course, used to describe an otherwise traditional child. In context, "furkids" can have brothers and sisters who are "skinkids", and vice versa -- but the two need not necessarily be related; neighbourhoods can, apparently, have any number of both.