From what I googled I didn't get an answer as to why Eggnog is considered a holiday drink. Why is it associated with Christmas? Is it still a holiday drink today?
The word nog was an Old English term for ale, and a noggin was the cup from whence it was drunk.
Although most Americans think of eggnog as something they get out of a milk carton during the two-week period leading up to Christmas, eggnog descends from sack posset, a strong, thick English beverage built upon eggs, milk and either a fortified wine (like Madeira) or ale. It was a highly alcoholic beverage, often served so thick it could be scooped. It was also very much an upper-class tipple, as rich folks were usually the only ones who could procure the proper ingredients.
My own two cents: Would you want an egg and dairy drink--hot or cold--in warm weather? Also, the tendency to get colds in winter may have meant it was made more often during cold months; and, the inclusion of mace and nutmeg is a natural fit for winter-spiced foods.
Eggnog appears in grocery stores in (at least) the US from late November through early January. Whether it's specifically a holiday drink or more generally a seasonal one is unclear.
According to the Wikipedia page, eggnog gained its holiday association in the 18th century in the US:
The drink crossed the Atlantic to the British colonies during the 18th century. Since brandy and wine were heavily taxed, rum from the Triangular Trade with the Caribbean was a cost-effective substitute. The inexpensive liquor, coupled with plentiful farm and dairy products, helped the drink become very popular in America. When the supply of rum to the newly founded United States was reduced as a consequence of the American Revolutionary War, Americans turned to domestic whiskey, and eventually bourbon in particular, as a substitute. Eggnog "became tied to the holidays" when it was adopted in the United States in the 1700s. Records show that the first US President, George Washington, "...served an eggnog-like drink to visitors" which included "...rye whiskey, rum, and sherry."
The same page notes that a hot version of the drink, Tom and Jerry, is mentioned in two 20th-century Christmas-themed works of fiction, Damon Runyon's 1932 short story "Dancing Dan's Christmas" and Yogi Yorgesson's 1949 novelty song "I Yust Go Nuts at Christmas".
There's nothing in the ingredients that is especially available during a particular time of year (unlike, say, apple cider), and yet it is not routinely available the rest of the year. I suspect this is due to combination of (a) a cultural association between eggnog and winter holidays and (b) the perception of a limited market for something with a limited shelf-life -- suppliers have decided that it won't be profitable year-round and instead promote it during a time of year that already has some traditions and anyway is more likely to have parties.
That said, you can make your own and enjoy it in June if you like.