This would almost be funny if it wasn't so horrific. I am fairly sure I'm the biological father in question, though many details have been changed by the OP. If I'm not, my experience may still be relevant.
Despite not being with my son's mother, I've been fully (and of course willingly) involved in his life since birth. She got together with her now-husband when my son was about one, though she didn't marry him until many years later. When he was around 3 or 4, I picked him up from nursery to discover that his name on his coat peg had been changed to her then-boyfriend's surname. Needless to say I got that reversed immediately, but since then she has carried out a systematic campaign of calling him by his now step-father's name while he's with her, especially since they had further children. She has put massive pressure on him by making it obvious that she wants his name to be changed - of course when she asks him he tells her what she wants to hear, which she then uses to try to justify changing it as she says it's what he wants and I'm being nasty and horrible for not letting him change it. What he tells me is of course completely different. A couple of years ago (he's older than the child in the OP) we jointly organised a birthday outing for him, and they got him a football top with his step-father's name on it (and no sign of my/his name at all). For his sake I didn't say a word about it despite being furious. This is just one example of many.
For the last year they've moved further away from me (entirely against my wishes), and I recently discovered that he had been enrolled in the new school with a double-barreled surname, with his step-father's name last. This was done entirely without my knowledge let alone permission. I don't feel I can contact the school and have it changed now as it would be too embarrassing for him, and the last thing I want to do is upset him, but what his mother has done is so wrong that it makes me incredibly angry and upset.
And for the record, I did offer to let the name be double-barreled, as long as my name was retained last. That was unacceptable to his mother, as it wouldn't allow her to quietly drop using my name at all, which is quite clearly what she really wants. The whole situation makes me sick and I don't really know what I can do about it, as regardless of the legal situation I have no control over what she calls him while he's with her, meanwhile my son is caught in the middle.
As to why I said it would almost be funny, if the OP is my son's mother, she clearly started making things up to paint me in as bad a light as possible once it became clear that the tide of opinion was against her. I am gratified that even this didn't entirely work.
So yes, YABU.