I know this thread is about families of beasts persay, but I would like to say rather than being atatched to a specific animal I am more atatched to skins of a certain time frame. I've found that older, not neccessarily unique pets are my most frequent companions.
Humar the pridelord will always be my constant cat companion. Shy-rotam is the whole reason I made a hunter and will always either be with me or in my stables. I never finished the quest just in case one day I might have to find her again.
Ironjaw will always be my constant wolf companion. I don't even want to think about taming another wolf.
I miss my old pink and purple terrokar owl. I miss hearing melee complain about him screeching their ears off in raids ^_^ Although I think to sate that indulgence I will be adding a monkey to my stables soon

I will say though as far as newer classes of pets, I love the shale spider. I have 2, and even though they only have 1 ability unlike their beast mastery only compatriots, if I'm going out into the open world you can be sure one will be with me. Amazing.
Blizzard has no doubt put out some amazing beasts, with lots of families having many beautiful skins but there will always be that one or few that wiggles its way into your heart regardless of what it is or what it looks like.