I think it all depends on if your ex is also involved with their friends, she is willing to accept that you refer yourself as the children's mother and most of all....your child is ok with it! For me the second two are the ones that prevent me from just being another mother to my children's friends. I understand to a point since my children do have a biological mother, but I have tried to tell them it has nothing to do with me trying or wanting to be there mother, its for convenience only.
I've had to go through this situation many times, as I have 3 children. My children do not mind me being referred to as their mom when out and about or if it is an acquaintances but to there friends and anyone that knows us as a family I am their dad. I understand they do not want to call me mom or refer to me as their mom, they have one, but certain times it would be nice so I wouldn't have to out myself always. It also such a waste at times cause most of their friends don't even meet their biological mom, often their friends that are told think my children are joking or lying to them. My middle child has referred to me as his mom early on in his friendships but then eventually tells them. It does work out better, sooner or later an awkward situation will arise. The same goes with telling someone that you may be in a relationship with....this is even harder for me to do!!
I do have many times that I am at my children's school events and am assumed as their mother, at least don't have to out myself at these and can enjoy my time being there as just another parent. I hope you have an understanding ex and she allows the children to have 2 moms....unfortunately I have an ex that is not one of the greatest mothers, but forbids my children to call or even think of me as a mother 🙁