Okay, Santa is huge. No, I’m not talking about his belly. Mythologically, he is many people. The name Santa Claus derives from the Dutch figure Sinterklaas. But stories of gift-givers are throughout Europe (and even other locations, but they aren’t attributed to being part of Santa). There is Saint Nicholas, Odin, Krampus (a kind of opposite who gives bad gifts and frightens people on the naughty list into behaving), Ded Moroz (from Russia), and still more in Austria, Hungary, with variations on the story.
The short of it is that people who do the right thing, who behave nicely throughout the year will be rewarded with a gift giving either in the form of presents, gifts in stockings, or even in shoes.
Now there are some remarkable similarities across the board with these gift-giving figures. First is that it always in secret. The give is not supposed to be seen. It is an altruistic anonymous gift where the giver does not want to be identified as having given someone a gift. Sheldon is somewhat right that gift-giving (at least in modern times) has sparked a kind of reciprocity. Ie: I gave you something, so now you must give me something. The anonymous nature of the gifting prevents this onus from happening.
Second is the appearance. Almost universally, the Santa figure is dressed in robes wearing a hat. The robe is most likely tied to the Greek St. Nicholas, who was of the clergy, and clerical robes would be his default garb. But in colder climes, the fur-trimmed robe also works as a means to stay warm in the inclement weather. The colors of red and white are not universal, but almost always some variation on red and white, green and white, purple and white, etc. occur for the figure.
The figure is always older, having a long, white beard. In most of the cases, the figure is overweight, though to varying degrees. The age and physical condition also lend themselves to the appearance of a staff or cane. Again, for Nicholas, this makes sense as a member of the clergy to carry a staff of office. Odin, too, was always seen walking around Midgard carrying a staff (which in reality was his spear Gungnir). Now, I’m not completely certain, so don’t quote me, but I think the candy cane is an evolution of the staff/cane carried by Santa.
What I like most about the Santa myth is how pervasive it is. It spread from multiple points across Europe, each one weaving together with the other to create a greater mythology as a whole. It’s nearly impossible to dismiss the figure entirely as there is so much corroboration from other cultures. This enabled Santa to achieve his worldwide reputation that extends past normal cultural boundaries. Santa doesn’t care which culture a person belongs to, merely that they are nice. So what was strictly a European figure, rooted in Christian, Scandinavian, and Celtic mythology becomes a universal mythological figure that has the power to adopt even more cultures into it.