...by the end of the 1920s, a standard American Santa—life-sized in a red, fur-trimmed suit—had emerged from the work of N. C. Wyeth, Norman Rockwell and other popular illustrators. In 1931 Haddon Sundblom began thirty-five years of Coca-Cola Santa advertisements that popularized and firmly established this Santa as an icon of contemporary commercial culture. http://www.stnicholascenter.org/Brix?pageID=35

The figures that approach the modern Santa with the fur hat come via Germany. Different parts of Germany had different present bringers. The North had Sankt Nicholaus, who was, and still is, dressed as a Bishop. As you can see, he is dressed in red, but no fur hat. In the south, figures such as the Weihnachtsmann (Christmas Man) or the ChristKindl ( Christ Child) deliever presents. The Weihnachtsmann is a little grimmer than friendly St Nicholaus and seems to derive from pre-Christian times. Sometimes he is called der Krampus (from the word for "claw"), and accompanies either St Nicholaus or the Christkindl, frightening small children and punishing those who are bad. The ChristKindl has transformed, in some areas, into a Weihnachtsmann, even though the name means Christ Child. The name was corrupted into Kris Kringle in the US. The Weihnachtsmann sometimes does wear a fur lined hat, but often as not a furlined hood. I found this interesting piece of information on a site: The figure of Santa Claus is based St. Nicholas, an actual Christian bishop who lived in the fourth century. Born in what is now part of Turkey, Nicholas was known for his deep sense of charity. Said to have been one of the most popular of the saints, Nicholas was named a saint for various works, including bringing miracles to poor families. Although representations of Santa Claus wearing white and blue clothing had appeared earlier, today's image of Santa in his red clothing and red hat apparently emerged in the 19th century, based on a popular Christmas card by a German illustrator named Schwindt. http://cp.c-ij.com/english/3D-papercraft/event/santa_hat_e.html I cannot find the original illustration, but it probably resembled this:(3) He is wearing the red fur lined cap.

The history of the images of Santa Claus are found all over the internet. As for the hats, they were and still are worn, for real.