Lots of beliefs are "well known" and nevertheless wrong. Just look at all the people, even in this thread, claiming that Santa Claus was invented by Coca Cola, even though a simple google image search will reveal illustrations of Santa from before Coco Cola was founded. As for the pagan traditions around new years they are not very well documented and a lot is speculation.
Of course gift-giving is known throughout history, but the figure of Santa Claus is associated with the Christian celebration. I'm pretty sure you wont find any reference to Santa Claus or Father Christmas from before Christianity!
So? Most religious traditions were stolen, whoop de do. What matters is how they're practiced in 2019, in which case Christmas is almost synonymous with Christianity.
> I'll just say that it's well known
There is no actual confidence in how all of this stuff came to be, just a lot of conjecture - and so no, it's not well known at all.
Christianity has no monopoly on Christmas.
Gift giving around the end of the year has been around since before 0 BC.
(And before someone screams "citation needed," I'll just say that it's well known, and if you didn't know it by now, there's a thing called Google and Wikipedia that you can go look at.)