The final lines are the textual evidence.
What a laugh it would have been
If Daddy had only seen
Mommy kissing Santa Claus last night!
This sentence, especially in the tone delivered, is constructed in an ironic way, as if the audience knows something the child speaker doesn’t.
That last part is also the answer to your question on why only this particular song is assumed to have an unreliable narrator. The narrator is explicitly a precocious child, sneaking around at night. who could easily see something they don’t actually understand. Most other songs about Santa come from either an adult or just some seemingly omniscient narrator, not a child participant.
There is also what others have mentioned–it doesn’t fit the mythos of Santa. Santa is happily married and very busy. He only arrives when all are asleep, including the adults. He doesn’t go around kissing parents. If he wasn’t perfectly good with not a single bit of naughtiness, he couldn’t be the judge of who who’s naught or nice. And it would become very creepy that he’s constantly watching kids to see if they misbehave.
The only real question in the song whether or not “Santa” is daddy or some other person in a Santa costume. The time period (as mentioned by others) is useful information. You didn’t cheekily joke about infidelity in the mainstream back in 1952, nor did you assume that mom was divorced. Plus there was already the known tradition of a father dressing like Santa for the kiddos (at least, as depicted in film), so you could buy that he just hadn’t taken his costume off before he was “underneath the mistletoe” and got a kiss.
Even assuming it was mom and dad getting some role playing kink on is a more modern interpretation that I believe was intended back then. Or, at least, if such was intended, it was deliberately delivered in a way to allow plausible deniability.
You specifically mention “Grandma Got Run Over by a Reindeer.” That song only has Santa doing Santa things–flying with his reindeer–and the person singing it is an adult who knows that most adults don’t believe in Santa. (Though, of course, it could have just been a normal reindeer, since nothing in the song explicitly says the reindeer was flying–it could have rared up to get the hoof prints on her forehead and the “Claus marks” could be anything. A skeptic would not accept the evidence given.)
I also note that the song discussing a dead grandma (despite the TV special editing that part out) suggests a sort of morbid humor that “Mommy Kissing Santa” lacks. Thus it being transgressive with a Santa who might do a hit-and-run is more fitting.