Fewer than one in ten Taiwanese is Christian, and December 25 is not a public holiday (it used to be until about a decade ago but not because it was Christmas, rather because it was the anniversary of the promulgating of the ROC Constitution in 1947). The retail sector obviously loves the idea of Christmas, and young people are getting into the habit of sending each other Christmas cards, but the festive season is still very much a minority activity. Most people save their money and energy for the Lunar New Year.
If you're thinking of coming to Taiwan at this time of year and worry about places being closed or packed with people, don't be. The weather can be a little cold, but all in all it's a good time to be touring the island.