The notion that Chanukah might correspond to the holidays of other cultural traditions is not a new one. The Gemara in Avoda Zarah (2b) speaks of an eight day holiday, Saturnalia, first celebrated by Adam when he noticed that the days had begun to get longer. Indeed, innumerable cultures hold celebrations around the winter solstice, and it is an intuitive annual landmark. In general, Jewish holidays do correspond to the seasons. Sukkot, a harvest festival, is not the same as Thanksgiving or Oktoberfest, but they are rooted in a similar consciousness (see here).
Christmas seems to have originated as the Christianized version of pagan midwinter festivals. Various yuletide customs have been traced (correctly or incorrectly) back to pagan practices. This being the case, an argument can be made (though not an especially compelling one) that Christmas and Chanukah share some lineage.
The historical question is largely beside the point, however. In the American cultural milieu, Christmas became a dominant cultural practice that excluded non-Christians (such as Jews and Chinese, who inevitably began celebrating Christmas together, with the former eating food prepared by the latter). Chanukah, by virtue of its temporal proximity to Christmas, naturally filled that void.
Not all “observances” and customs associated with this syncretistic celebration are as over-the-top as the “Hanukkah bush” or “Hanukkah Harry”. The giving of “Chanukah presents” has penetrated even the more traditional segments of the Jewish community. American gentile culture clearly associates the two holidays since it has adopted the generic “happy holiday” greeting to cover all observances associated with the season. Television commercials with Christian and Jewish symbols appear during the season. The term “Christmakwanzukkah” has come to symbolize this amalgam of various traditions and observances.
There has been a more recent shift in American culture for Jews to take pride in their identity, but this trend nevertheless uses Christmas as the yardstick. To wit: “Instead of one day of presents, we’ve got eight crazy nights”.