I understand what someone means by 'Christian name' and wouldn't dream of correcting them if they used the term, but it's not one I use myself, and don't think it should be used in schools or by public bodies.
I was taught at school that my Christian name was Doodle and my surname was Dog, and it didn't cross my young mind that Jewish or Muslim (or whatever else) children might feel 'othered' by this. They will have done, though, just as I would be nonplussed by being told I had a Muslim name.
'Given name' and 'surname' (or 'family name') mean the same, and aren't taking away from any religious connotations that they may or may not have. I was christened, but am not religious myself, so there is that to consider, too. Not everyone who has a name that was given to them in a religious ceremony as a baby sees it in that way in adulthood.