In this case, origins have little relevance. An archeological argument whereby we try to deduce what historical culture has a legitimate claim to a specific holiday is fallacious because it suggests a normative claim that only people who practice the religion of said culture can celebrate that holiday.
This is a bad point. Originally, December 25th was a celebration of the Winter Solstice, called Yule, for many European Pagans; when Christians began crusading and changing the Pagan ways of the indigenous peoples of Europe, they convinced the Pagans that their holiday, Christmas, was equivalent to Yule and, probably at least somewhat violently, changed the culture to Christianity over time, replacing many instances of Yule with Christmas. The Yule log and many other Yuletime traditions live on as "Christian" traditions; the Christmas tree is the "Christian" Yule log. Yule singing is called caroling now, as we also know.
My point is that your "normative claim" shit is irrelevant here because
the Pagans in Europe had this holiday first. Now, granted, the blood that was shed has been shed and we can't take that back anymore, but it really was not originally a Christian holiday that we celebrate in December. Jesus was probably not even born on December 25th.
If we agree that what makes people who they are is religion
You're not going to get me, nor many others, to agree on this one, so you may want to amend your argument to something less specious. Also, the rest of your argument has little-to-nothing to do with the holiday argument/discussion that arose, so I'm struggling to figure out what the fuck you're even talking about here. The point is that,
historically, Pagans celebrated "Christmas"
before Christians, though it was called something else and changed (probably violently, as Christians in the middle ages were wont to resort to violence, ironically) over time following years, probably decades or centuries, of religious syncretism.
Note that if your point is that "nowadays December 25th is Christmas so you're wrong because only the present matters and we shouldn't care about history or sweeping cultural movements and/or religious and cultural injustices", not only is that fairly blind of you, but it's also beside the original point I made.
So, basically, I have no idea what the hell you're trying to say.
edit: I am proud Dakryn agrees with me on this, no matter how much it may weaken my point. Look up "syncretism" on Wikipedia for more information.