My mother calls me mommy, but she know who I am?

Go with the flow, don't correct her.
I can not tell you how often I look at my grandkids and stop myself from calling them by their mom or uncle's name. They look so much like they did when their mom and brother were that age. I know who they are! it is just sometimes my brain looks at my granddaughter and it sees her mom. My grandson is the spitting image of his uncle.
I bet you look very much like your grandma did when she was your age. Take it as a compliment and move on with the conversation.
Also another thought...
Mommy's take care of someone that needs taking care of. Your mom needs taking care of so in a way you are playing a mommy role that might be where your mom's thinking process is.

My mom used to introduce me to the staff, as her mom, her sister Nan , or her little girl. Staff and I would just smile..

I spent a period of time trying to figure out what my mom did and did not know - at one point she could identify me by name and even as her daughter yet also said she didn't think she had ever been married or have children 🤔 When she went through a period of calling for her mother it broke my heart because I thought of that small inner child looking for the comfort of a mother, I think it made sense because I did care for her in the way a mother cares for a very small child. I eventually decided it was better for me to to stop trying to understand and just accept where she was each moment, but there were parts of her that were very lucid right up to the end.

My Mom occasionally knows who I am, sometimes she doesn't. And it can vary in the space of a minute. BUT, she always knows me as the nice lady who cares for her.
The nicest compliment I've ever had was when she was lost in her mind and I was trying to get her back to our shared reality and she whispered: "you're always so tender with me".

You're the one who cares for her and at any given moment she might not be able to access your name....So, mommy.