Nope, sorry.
I'm mom to my 3 kids-- and, no, I didn't give birth to my son.
But they're my kids. My dog is my pet. There's a world of difference in my book.
And I mean this with the utmost respect, but somehow it saddens me to think that your stepchild and your pet seem to be on the same level somehow.
I have no step kids. But I would love to see the prefix "step" permanently removed from society's vocabulary.
I agree. In my mind, the OP is much more of a parent/mother to her stepdaughter than she is to her pet. In fact, OP, I would have wished you "Happy Mother's Day" not because you have a pet but because you are a mother to your daughter.
I would not do it. I'm a mother and a pet owner, and I consider those two separate things. To compare the sacrifices I go through to raise my children to be productive, happy adults to taking care of my dogs kind of offends me.
Right. We have 3 children & 3 pets. We own the pets. We don't own our children. The two things are completely different, & the experiences are completely different. While I love our pets (the dog moreso than the cats), the love I have for my children is not even close to the same thing.
Comparing being an owner of a pet to being a parent of a child is a bit offensive to those who actually parent a child.
DD just had a baby after several years of having two dogs. She told me on Mother's Day,
"Wow, it really isn't the same, is it?" I told her, "Nope, but it's hard to understand."
I adore dogs and have always had at least one and sometimes more. The only reason my dog knows me as "Mom" is because that's what my dd calls me.

Our dog is WAY easier than raising a child. She always thinks I'm awesome and I certainly cannot say that for my dd!
Before we had our first child, DH & I had a cat, & I thought of Allie (our cat) as my baby. So I get it. While I never received a Mother's Day gift or card from Allie (from my DH), DH did get me Christmas presents from Allie. One year it was a little cat statue, & he dipped one of Allie's paws in bright blue paint to sign a card. (Blue paint everywhere! LOL!) One year, Allie got my DH a cat storybook for Christmas.
After we had our first child, gifts from the cat stopped - although we do still get our pets Christmas gifts.
When you have that first child, your perspective changes. And no matter how much you love your pets, you realize it's not the same thing at all.
Mother's Day is a difficult holiday for many for lots of different reasons. When we were trying to get pregnant & having some infertility issues, it was for me as well. While I loved our cat & considered her my baby, I never really thought of myself as her mother & didn't feel "honored" on Mother's Day because I had a cat.
OP, I'm not diminishing your love for your dog. I think pets do add so much to people's lives. I can't imagine our family w/o our pets. However, in answer to your question, no, I would not have wished you a "Happy Mother's Day" because you have a dog.