TL;DR: The premise is backwards. Pumpkin pie has a longer historical association with Thanksgiving, though it was associated in the northern U.S. with the winter months in general. This is due to both pumpkin pie and Thanksgiving having a strong regional association with New England until after the Civil War. It was only in the late 19th century that a specific association of pumpkin pie with Christmas began to emerge. So any association you notice of pumpkin pie and Thanksgiving is the more historic one, and not the dish's association with Christmas.

Long answer:

Your premise is a little bit backward. First, I think you may be mis-reading this because pumpkin pie is quite a common sight throughout the holiday season. Grocery stores keep ready-made pumpkin pies stocked from November through New Year's. Christmas Day is also "National Pumpkin Pie Day". At least as early as 1802-03, pumpkin pie was being associated with other celebrations during the holiday season aside from Thanksgiving. On New Year's Day that year, New England diarist Martha Ballard recorded in her diary that she was spending the day "at home...ma[king] pumpkin and apple pies".

Certainly since the late 19th century, pumpkin pie has had a strong association with Christmas. As some examples, the New York newspaper the Evening World reported on December 22, 1888, that they were holding a Christmas dinner for their newsboys, which would include "Christmas mince and pumpkin pies". An 1897 edition of the New York Tribune ran an article under the headline "Christmas Pumpkin Pie" which was reprinted in the Virginia Enterprise and gave cooking tips on the annual traditional dish. In 1910, the Denison Review newspaper in Denison, Iowa, contained the Christmas Day menu for the local Hotel Denison, which included among the offerings "Christmas Pumpkin Pie". A September 11, 1919, article entitled "Pumpkin Pie Always Good" published in the Londonderry Sifter in South Londonderry, Vermont, made the claim:

"...[W]e are creatures of habit, and are inclined to follow traditions in our cooking. Our foremothers made pumpkin pies in the fall because the pumpkins are ripe at that time. She continued the pies through the cold weather as long as she could keep pumpkins in her cool cellar. Thanksgiving and Christmas dinners always included pumpkin pie. The pumpkin supply was exhausted about this time, and therefore no more pumpkin pies were possible until the next autumn."

The article goes on to say that "times have changed" and that pumpkin pie can be enjoyed throughout the year due to modern canning technology.

In 1929, the Midland Journal newspaper in Rising Sun, Maryland, ran a poem under the title "The Christmas Pie" which started with the verse:

"O Tommy's mother made a pie,

A Christmas pie she cooked,

And Tommy thought that pie the best

At which he ever looked

It was of pumpkin yellow,

And gingered for some pep,

And Tommy though that that big pie

Was too good to be kept."

And the third verse:

"But he heard his mother coming

And with a stifled cry,

He turned to run but tripped and fell,

Right in the pumpkin pie,

Arrived the Christmas dinner,

A different pumpkin pie.

But where was Tommy's portion?

He had no pumpkin pie."

However, there is some truth to a stronger connection of pumpkin pie to Thanksgiving than to Christmas, but it isn't because pumpkin pie "went" from being a Christmas staple to being a Thanksgiving staple. It was the other way around: it was associated with Thanksgiving and then eventually started to be associated with Christmas, too. In the Christmas Eve 1952 edition of the Key West Citizen in Key West, Florida, an article under the title "America's Christmas Based On Old European Tradition" reported:

"The first Christmas menu in this country, that of the Puritans aboard the Mayflower, consisted of: salt fish, bacon, brussels sprouts, gooseberry tarts, and plum pudding. Quite a difference from today's Christmas repasts! It is in this category that America can lay practically her only claim to fame, namely, the Christmas turkey and pumpkin pie. However, there are some who would dispute our claim—they say that it's just a leftover from Thanksgiving Day."

This is likely because, more than pumpkin pie being associated with a specific holiday season celebration, it was associated with a specific region in the United States. And that region was New England.

A brief history of the regional association of pumpkin pie and New England can be found in the book America's Founding Food: The Story of New England Cooking by Keith Stavely and Kathleen Fitzgerald. As the book reports, it may not necessarily have started out as a regional dish in the 17th century, but by the early 19th century, it was associated with New England and the North. Pumpkin pie was brought over from England at the time of Puritan colonization. It was considered something of a poor man's food, and only baked when better fruits like apples and berries couldn't be used. In the early years of colonial New England, however, pumpkins were much more available during the winter months, so it started to become a traditional dish there, so much so that by the 1760s, even upper-class Bostonians were celebrating during the winter months by happily eating pumpkin pie. The first American cookbook, entitled American Cookery, written by Amelia Simmons and published in Hartford, Connecticut, in 1796, includes two recipes for pumpkin pie. An 1831 cookbook called pumpkin pie one of the defining characteristics of the "universal Yankee nation." An 1839 cookbook called it a "real yankee pie".

An article in the July 6, 1868, edition of the New York Tribune entitled "Our Restaurants" detailed some of the cuisine options in New York City, including those serving traditional regional American cuisine, of which the article stated:

"If you are a New-Englander, you find here baked beans, succotash, New-England puddings, and the ever-famous pumpkin pie. If you are a Southerner, with unreconstructed palate, there is any quantity of corn bread and bacon to sate it. If a representative of the Great West, you can get plenty of beef, pork, and vegetable growths raised on your own soil, beside which are common to the thirty-seven States."

The March 3, 1864, edition of the New York Post described the New England House, a restaurant in Brooklyn specializing in the cuisine of New York's northern neighbors, though it served local New York dishes as well:

"The pork-and-beans—solid Yankee dish, beloved of New Englanders—the pumpkin pies, the apple-sauce, the doughnuts, the cider, are all of the best quality, and moreover, are clean as well as abundant."

Pumpkin pie's association with New England is important because, in New England until the mid-19th century, Christmas wasn't really celebrated, but Thanksgiving was. Thanksgiving was an important annual holiday in New England before the Civil War, celebrated around the end of November, while elsewhere in the United States, Thanksgiving was only celebrated sporadically (typically when a governor made a declaration for a day of thanksgiving) that wasn't necessarily confined to the winter season.

This is discussed in the books The Battle For Christmas by Stephen Nissenbaum, and Christmas In America: A History by Penne L. Restad, as well as in the journal article "The Making of the Domestic Occasion: The History of Thanksgiving in the United States" by Elizabeth Pleck and published in the Journal of Social History. The Puritans had considered Christmas to be a "papist" or Catholic invention that did not have any evidence for it in scripture, and they chided the Anglican church for continuing the celebrations. From 1620, Christmas celebrations in New England were socially unacceptable; in 1621, governor William Bradford caught a couple of people celebrating by playing outdoors, and Bradford ordered them to get back to work. Between 1659 and 1681, celebrating Christmas in Massachusetts was outright illegal, with the penalty being a fine of five shillings.

While the penalty went away, Christmas and the holiday season apart from Thanksgiving was largely ignored by Puritans in New England during the 18th century as well. In 1790, the U.S. capital was in New York City, where Christmas and New Year's were publicly celebrated, and the New England-born and bred Second Lady, Abigail Adams, wrote in her diary at her surprise at how different New Year's was than in Massachusetts where it wasn't really celebrated at all.

(cont'd...)