I once taught kindergarten students. Because they spent so much time with me - they would sometime want to call me "mommy". I would correct them and gently ask them to use my name. But children, simply call those who are "good caregiver", especially if they have not been coached, "mommy whatever" I seen it happen so many time.
As a stepmother, I insisted however that my stepchildren refrained from calling me "mommy whatever" because I knew that there mother would resent it. Actually after many years of marriage- we have all resolved our insecurities. All of our children have grown to love each mommy. It takes time. But it makes for a better life when kids don't have to choose but can just relax and enjoy life. This mother has insecurities and issues- as anyone would so brutally betrayed. However, laying a guilt trip on a small child about something so nature as calling a caregiver mommy is pretty sad. Clearly, the wounds are still fresh for this ex-wife. When they begin to heal she will see she has bigger fish to fry and focus on her children understanding that her problem with the new family is not theirs.