I think that you and your mother should make an effort to discourage your daughter from dog role playing.
Not all role play, it is an important developmental stage, just don't actively encourage her to be a dog. To ME (an Australian with European heritage) it is cute and 'normal', but to the mother of your child, a woman I am assuming you (even if not right now) loved and cared about, it is highly offensive.
I'm making some assumptions here about your relationship with her, so do correct me if I'm wrong. I'm assuming you chose to have a relationship and start a family with a woman from a significantly different culture - you need to accept there will be repercussions to that and you will need to make adjustments to support her - regardless of how irrational her cultural beliefs seem to you and your mum.
The thing is, to an Australian acting like a dog is 'no big deal', but to her it is. If its 'no big deal' then it can also be no big deal to just stop encouraging it. There are thousands of other animals she can pretend to play as.
We can all assure you that pretending to be a dog is developmentally 'normal' as can experts etc. You can dig your heels in and insist that you are right, she is wrong, and that she must accept this fact - but at what cost???
Or you could be sensitive to her cultural background and give on this one "little thing" (and yes, there may be others down the track) but sometimes we need to do things just to keep people close to you happy. Is it really any skin off your nose if your daughter only gets to pretend to be rabbits or chickens and not dogs? But sticking to your guns has possibly already ruined your relationship and now it is potentially going to lose you your child.