My step-daughter calls me Mom and its okay by me and her dad, but her birth Mother has a issue? HELP?

My stepdaughter is 6 years old now and i have been with her father since she was 3 years old. It wasn't until about 1.5 year ago she started calling me mom but i never corrected her because i didn't ask her to call me that.We have sole custody and alway have, Infact she asked me if she could call me mom. When i asked why she decided she wanted to call me that she said its because she lives with us and i do everything a mommy does for her like cooking,cleaning, and "girl stuff" :) and i taught her how to read,write,tie her shoes,swim,monkey bars plus our most recent daughter calls me mommy and she likes calling me that (her words) , but everytime her birth mother calls her which is rarely...or sees her which is even more rare... she gets upset and snaps at her if she hears her call me mom.... and i think it is confusing her. I dont know how to handle the situation..... need opinions? How do i handle this situation i dont want her to feel like she can't call me mom, like I dont accept her as my daughter cuz i always have....?