To answer the first part of her question, yes, I do call myself a dog mom. My daughter does too, and always buys me cards from her and from the dogs. So while I have a human daughter, I also have kids of the four-legged variety. We compete with our collies at dog shows, and have traveled 16 - 20 hours by car with our collies to reach some of these shows. I know many moms of human children who travel these distances, to take their human kids to different competitions they have entered.
I take my collies to their doctor when they are sick, I buy them toys at Christmas and on their birthdays. (Which we wrap in pretty paper) I talk to them, sing to them, hug them and have pictures of my collies hug up at home and at my desk at work. (Right next to pictures of my human daughter.). I feel pride when my collies win at dog shows, and I worry when they are sick. So yes, I am definitely a dog mom. And I'm proud to say my daughter feels the same way about our dogs.
The second part of her question, well that's a little harder to answer. As I am one of the few dog parents where I work, I tend to not refer to myself as a dog mom there. I know it is a term that would cause eye rolls, and possible negative comments, so I avoid the subject at my place of employment. But I avoid it because it's where I work, I don't care what strangers think, and I would use the term without hesitation elsewhere. Because to me, the definition of a parent is one who provides care and love, who guides and educates, throughout another being's lifetime. And isn't that what we do for our dogs? So who else is a Dog mom or Dog dad?