HCP working in a paediatric subspecialty.
I feel your pain. It would drive me totally potty.
However practically at work I have less than 10minutes to see, assess, build a rapport and explain a medical procedure to each child and learning parents names sadly gets squeezed out. There is also a phenomenon when I tell people my name "hello my name is" and people never reciprocate. There are a few families I see recurrently (my specialty isn't big on continuity!) and I still don't know the parents names. It isn't always easy to quickly access that info in our notes either. One ward at work piloted parent's wearing name badges. I liked it but I don't think the parents did as it has stopped.
I usually start consultations with "and who have you brought with you today" to the kid indicating the adults. The pointing adaptation was necessary as I have had huge long lists of toys names before!! And kids rarely tell you their parents first names!
I also like to clarify who is who as it helps sort out who can actually consent for a procedure e.g. Not normally A grandmother, or mother's partner and guessing family relationships leads you down a rabbit hole. I have made so many cock ups about family relationships over the years that I try very hard not to guess - asking a mother if she is the grandparent NEVER goes down well. Or guessing that the Mother's boyfriend is the Grandfather is similarly bad. Worst ever was when one of my nursing colleague got generation & gender wrong eg. Mistook the child's aunt for a grandfather.... anyway, I digress.
My compromise wrt to "the Mum issue" is I never use a relationship title without a personal pronoun (e.g. Your Mum or Tom's Mum). This does lead me into some slightly odd sentence construction but I cope!
I tend to talk primarily to the children which also helps me avoid calling the accompanying adults anything. Only time I wouldn't! So this is if they are seriously tiny (baby) or very very significantly developmentally delayed. This is also part of rapport building and I rely on parents chipping in if the story goes wildly off track... Sometimes I can spot it but not always! "I broke my arm falling off the space rocket" will trigger more parental clarification than "I fell out of my bunk bed" although neither story might actually be true.
As a further aside - Letting children speak to HCPs helps them learn an important life skill and if your child is in secondary school parents and you still routinely answer all the questions in a consultation even if directed at your kid I would ask you to question why you are doing this. I know some kids just aren't able/ confident enough etc. but I really appreciate parents who at least leave their child a pause to potentially answer in.
I also challenge all my trainees and the junior nurses to think about this practice and the children's hospital I work in actually has special communication training for staff that covers this point.
So I don't think you will get staff remembering your name especially in healthcare. I can't talk about education. But people should be able to call you X's Mum instead which while a bit impersonal & failing to recognize a parents personal identity separate from their parent role sn't technically incorrect.
This phrasing grates on my sensibilities a lot less.... but I'm interested in parental feedback.