I have been in a similar situation, and I have watched step-siblings go through it, too. I know you don't intend to, but, by constantly having these conversations with your step-son, you are creating pressure on him. I remember the day my step-brother (I usually refer to him as my brother.) asked my Mom (and me) if it would be okay if he called her Mom and told everyone she was his Mom. It added a new level of comfort to the blended family. He doesn't have contact (by his own choice) with his bio-Mom, though, so that was different for him; he was filling a gap, to some extent, of his own volition. My step-sister began referring to my Mom as Mom but only at home, which was fine. (My step-dad had full custody.) She seemed very torn about it, and I think she did it to fit in. It caused trouble between her and her Mom, but she continued doing it, because she later said she didn't want to upset anyone. It created a scenario where she seemed to live a double-life depending on who she was with, although my Mom and step-Dad repeatedly told her it was okay to do whatever she felt comfortable with.
My Dad remarried when I older, and his wife started introducing me as their daughter", which upset me horribly. I felt that she was taking credit for someone else's work; when I discussed it with them she got upset and it caused tension that took years to resolve.
Maybe you could tell your step-son that you love him regardless, because the title isn't important. Ask him what his feelings are and then just listen without interrupting. If he is interested in picking out a special name for you, then maybe try using the internet to look up words/names that mean Mom, etc. in other languages.
I would not encourage him to do anything behind his Mother's back; I feel that it is a parent's job to teach children to respect their parents and others, and lying by omission is not a pattern you want to create. And, if the main issue has arisen because your daughter was calling you by your first name after hearing your step-son, then maybe this would be a good time to begin teaching your daughter what is appropriate for some if not appropriate for all. i.e. You wouldn't let her eat bread if she had a wheat allergy, even if your step-son was eating bread, because there are different circumstances; same thing with the name situation.
I hope this helps; good luck, and kudos for being a good Mom and step-Mom.