I'm 27 and my step dc are 16 and 12. I have been told by both a family member and a colleague I shouldn't let them cal me 'Mum' as it's inappropriate/ embarrassing in public as we clearly look our ages (well possibly younger in my case, often a bit older in 16 yr old dd's case).
I am right in thinking it would be incredibly cold of me to tell them to stop? I'd fell like I was pushing them out of the family unit. I have acted as a mother to them now for years. They started off calling me by my name but dd in particular disliked this as it caused questions from her friends (who were at the time too young to judge my age) and asked to call me Mum. DS being younger called me Mum quite naturally from a young age as it went with 'Dad'. He kept slipping up and saying 'Mum', then gave up correcting himself. The title stuck. I do everything for them a birth mother would, despite being too young to have actually given birth to them. It does get quite a few looks on the bus or when they shout it out in the Supermarker (hv eyebrows nearly popped off the first time and their teachers have looked like they're burning to ask more at parents eve, I know it must be gossiped about) but I just brazen it out and I never explain they're not 'mine' as they always look shy and awkward if the topic comes up, plus I don't want them to make them feel less to me than ds I gave birth to.
They call their birth Mother 'Mama' btw, and they have a good (but very different) relationship with both of us. If either of them ask to call me by my name again in public or at home I'll have no issue with it, as they may do as they become more aware of the age issue. Though if dd hasn't stopped saying Mum by 16 I doubt she will. I think it's a bit like giving a term of respect to me.