I noticed that the Stackexchange sites are having a Winter Bash, and one of my friends recieved a hat on MSE. Why isn't this on MO? It'll be fun to have something like that here, wouldn't it? (I suspect this is because MO doesn't have a ".stackexchange.com" in its address, but I could be wrong.)
Somewhat before the start of the Winter Bash, the SE team contacts the per-site moderators to ask if the site wants to participate in the event or not (this year the default assumption being that the site participates, differing from last year's practice).
The reason that MO does not participate is (almost surely) that its moderators decided the site does not want to participate and communicated this decision to SE. Note that also last year MO did not participate for reasons along the lines given above. This has nothing to do with the domain name; stackoverflow.com has hats while a site with URL [something].stackexchange.com domain might not have them (I did not search for an example, but as explained at the start it would be possible).
To sum this up: on the technical side there is no issue at all, MO could have hats; it was simply decided against having them for this year.
I'll just add that quid is correct. The moderators were asked in 2013 whether MO wanted to participate in Winter Bash, and we declined. We thought it would not be wise, as it would be sure to turn off some of our esteemed users who do not come here for silly fun times. We were invited again this year by the StackExchange team, who essentially asked "we don't suppose you've changed your minds, have you?" and we of course said 'no'.
Some of the comments below the question seem to corroborate our suspicions which led to our decision.
I like the winter Bash, and several years ago I was even a champion in terms of numbers of hats in the TCS sister site. But I can understand that some mathematicians do not share my enthusiasm. It took decades to have an analog of Nobel prize in mathematics so it may take decades (if at all) for winter bash to be embraced on MO.