It has a function if you operate as a couple making it clear who is with whom - useful in certain circumstances just as at Primary School I was only ever "X's Mum" . My own surname - different from my kids' - was of no help in identifying me at all.
But you don't operate as a couple when it comes to your own birthday, unless you both share the same birthday and you like receiving joint cards and presents, maybe. it might be slightly more understandable in the context of a wedding card, especially if from somebody who knows your new husband but not really you.
In the context of the school, somebody referring to you as Katie Jones' Mum makes perfect sense; but it would be very weird if they insisted on calling you Katie Jones Sr, though, wouldn't it? There's nothing wrong with referring to you as Phil's wife, Janet's sister or Emma's husband if they only know of your existence in the context of Phil, Janet or Emma, whom they do know well.
e.g.:
"Oh, hi, you're Phil's wife, aren't you?"
"Yes, that's right - I'm Jill. Nice to meet you."
"Nice to meet you too, Phil's wife."
The first two lines of conversation are perfectly normal; the third is weird and rather rude, to say the least.
And then, if your dentist, say, happened to know your daughter's name, because she's also one of their patients, and then proceeded to send you appointments for your own check-ups addressed to Katie Jones Sr, even though they know that your own name is Caroline Peters. Wouldn't that at least irk you? That's pretty much the equivalent of this scenario.