For a long time, I resisted the “Daddy” thing with Iiko Puppy-Dog Boy. I have since acquired the habit. Now my parents are Grandmom and Granddad, and their dog is “your little sister.” Oddly, I am “Uncle____” to my parents’ dog.
I sometimes say that Iiko is “the small, dull child I may never have.” I say this even though he’s whip smart (not that it does me any good) and weighs +/- 80 lbs.
Chrome
I refer to myself in the first person. With my boyfriend’s cats, I refer to myself as “I” - they don’t like it when we patronize them. 
Speaking just for myself, the mommy/daddy thing creeps me out. They do that at the vet–when they give the cat back, they’re like, “Here, go to Mommy!” It gives me the willies. I always want to correct them–“Please, we don’t do that!”–but what good would it do?
On the other hand, I’m not above asking the cat, “Who’s my baaaaby?” so I guess I have no high ground to stand on.
Just a few weeks ago, I remember asking someone if they were their pet’s “big sister” or “mama.”
I had dogs when I was a little boy, and there was no familial metaphor. Like Crunchy Frog said, I was “the boy” and Blackie (and later Jimbo) was “the dog.” My dog now is variously “the boy,” “the little man,” “Bubba,” or “shutthefuckup,” but I am always “Daddy.”
The girl is the girl, the boy is the boy, I am dad, and the wife is mom. Unless the dog screws up, which is quite often. The I become The Lord Your God and the wife turns into the Angel of Death or Jason, depending on how pissed she is.
I am mommy; hubby is daddy; daughters are sisters to Buddy dog. Collectively we refer to ourselves as his “pack.”
Maybe if I’d gotten my cats as kittens?
We’re more like roomates. I’ll be lying on the couch, saying, “dude, stop eating my cheetos. You just want to lick them anyway. Gross!”
I’m the smart but boring one who’s always trying to settle everyone down. But I have the money for catnip, so they let me hang out.
They’re the cute but irresponsible ones who just party all day 
I’ve never done the “mommy” thing. Though I did introduce myself, at a wedding, as “Sid’s mom”. [sub]I’m not completely crazy, it was to someone who had not met me but had met my cat. He’s very outgoing, and liked to visit the other people in my building. I told you they’re the outgoing ones…[/sup]
~~ Herr, Johann Gambolputty de von Austern schplenden schlitter
crasscrenbon fried digger dingle dangle dongle dungle burstein
von knacker thrasher apple banger horowitz ticolensic
grander knotty spelltinkle grandlich grumblemeyer spelterwasser
kurstlich himbleenisen bahnwagen gutenabend bitte ein
nurnburger bratwustle gersputen mitzweimache luber hundsfut
gumberaber shonendanker kalbsfleisch mittler aucher von
Hautkopft of Ulm…
…for both dogs and cats… thanks for asking!!!
Oh, and no, somehow I never refer to myself to my dog. Odd, that.
My sister and I kid around about it, “Pat my niece!”, but I’ve never called myself mom with my critters.
Make no mistake, I love my dog unconditionally because that’s the way she gives love back. I scratch her silly ears, croon “Brown Eyed Girl”, romp, cry, talk, etc. and she cocks said ears, nudges me when she wants a walk, plays, has snits, snuggles, romps some more…
She’s a miracle. She’s a DOG, a distinct species which somehow bonded with humans. She isn’t my offspring. She’s a DOG, a full-fledged, noble, hard-wired canine who nonetheless accepted me as pack.
She’s herself, a dog. It’s a chosen thing, and a gift. A HUGE gift. Okay, so I sing (badly) to her, tell her she’s boooful (and she is)…aw, shit. I’m just grateful, and enriched.
Veb
I have two cats. One of them I raised from a teensy little guy into the 17 pound whopping monster that he is. He gets referred to as my son, and I’m his mommy. The other cat I got as a mostly-adult shelter discovery, and I’m not her mommy at all. I’m her person, mostly. “Come to your person, Kopushkin.”
Of course, she’s been really bad sick, and is currently recuperating at the Corr-friend’s house, so I refer to HIM as her boyfriend. (She’s a total flirt to him, and adores him.) He calls her his other girlfriend. She eats that up.
sigh I knew I wasn’t her real person, but was taking care of her until that person came along, though. It’s been seven years, but she’s totally attached to him. Time will tell, I guess.
Corr
My girlfriend refers to herself as “Mama” or “The Mama”(as per an old cartoon named “Dinosaurs”, I think?), and to me as “Daddy” or “The Daddy”. I’ve gradually started doing this as well, although I never did it with any prior pets.
The dog herself, a spoiled and ludicrously silly chihuahua, is “The Puppy”, “ChiZilla”, “The Dog”, “PupSicle”, “PupStrosity”, “The Baby”, “PupplUpagus”, or just “Silly Dog” when not being addressed by her proper name “Chili Dog”.
She is also “Chili Dog dog” to me, since previous pets such as Oreo and Mittens and Smudge would be addressed as “Oreo Kitty”, “Mitty Kitty”, “Smudge Kitty”, etc, so consistency + a little silliness makes her therefore “Chili Dog dog”. Or when she’s exceptionally in chracter, , “Chili Chili Dog dog”.
I can’t say that I’ve ever referred to myself in the third person when talking to my pets or any of my human friends (after all, I’m not Bob Dole). Usually, it’s “I” as in “I said Sit!” However, I will admit to occasionally calling the collected animals of the house “the kids.” Still, the whole Mommy/Daddy, and especially the Grandma thing has always kind of disturbed me. If one of my coworkers or the owner of one of our patients is into that sort of thing, I’ll usually go along, just because the animal’s used to it and finds it relaxing.
I always use the first person when talking to the dog. I also try to reason with him, share my feeling with him, and have been known to lecture him when he does something wrong. So perhaps I show an unusual amount of deference to him. Or perhaps I’m just pathetic and lonely. Given that it’s a Saturday night and I’m sitting at the computer, that second one is looking pretty likely.
I refer to my self as “Daddy” when talking to the dog,but I use the first person when talking to the cat(Not that the cat listens).
“Me” or “I” usually. I hate the mommy/daddy thing. I’ve had many arguments with my mother when she insists on calling me one of my pets (a dog and a ferret) mommy. They’re not my babies. My buddies, yes, but not my babies.
I’m Mama, and my husband is Daddy.
I call my dogs my Smelly Kids, as in “Where are all my smelly kids? You’re happy to see Mama? You love your Mama? Your Mama loves you! Your Mama loves you!”
Each dog has their own pet name too. Pogo (the smallest) is Little Man, Herbie (medium) is Fat Boy or Fat Sad Boy (he’s a hound and has a naturally pathetic look on his face) and Francesca (the largest) is Big Bitch.
Oh, and my Father In Law is their Gammpy and my Mother In Law is Grammy.
Rose
With dogs, I tend to refer to myself as the alpha (too many years training dogs). tdc refers to himself or his family as the animal’s people (“We’re Penny’s people”), which I think is cute.
But, then again, I would.
I should add that on a basset hound list I read, most people are referred to (from the pet point of view) as “Food Slaves.” That’s quite fitting. The Female Food Slave, and the Male Food Slave.
This thread is making me remember that Onion headline: NATION’S DOG OWNERS DEMAND TO KNOW “WHO’S A GOOD BOY.”
Well my dogs are called Lou-baby, the queen bitch, Buckie-boy, my big ol’ hunk of puppy love, and Jake-baby my snugglebuddy sugar bear. Sickening huh?
I’m Mom, my hubby is Dad of course.
Gosh, I m so ready for grandbabies.
I never mention myself with pets.
It’s only other people’s pets I meet, and I only have to “talk” to them when the jump me. So I think the only words I’ve said to pets this year is “Shoo”. And to the owners, “I don’t think it’s so funny! Will you get your animal off me?!”