the evidence for the two is pretty similar namely a temporal relationship.
Goyim made loaves with a cross marked on them around Easter time first then we made loaves with a key marked on them around PEsach (lehavdil!)
Goyim gave presents christmas, we give presents Chanukah (lehavdil!)
The only differecne is that presents is more recent. And as I said over there Give it(i.e. presents) a few hundred years, as it enters minhagim seforim and reasons are given eg the wrapped present represents the hidden pach shemen or some clever gematriya, or that Matanah stands for Mitzvos, Torah, nisism and Hashem which is what really won the war agaisnt the yevanim. At that time our descendants will have the exact discussion regarding presents as we did regarding the “ancient minhag” of shlissel Chalah