The winged wheel logo was actually borrowed from a cycling club in Montreal. Here's how it happened.

Detroit was awarded an NHL franchise in 1926, and the owners stocked their team by purchasing the roster of the Victoria Cougars of the Western Hockey League. The team was known as the Detroit Cougars for a couple of years, then the Falcons. They did not win often under either name.

In 1932, a millionaire named James Norris bought the team. Here's the rest of the story, as told by Bob Duff in his Detroit Red Wings article in Total Hockey: The Official Encyclopedia of the National Hockey League:

"Norris had been a member of the Montreal Amateur Athletic Association, a sporting club with cycling roots. The MAAA's teams were known by their club emblem and these Winged Wheelers were the first winners of the Stanley Cup in 1893. Norris decided that a version of their logo was perfect for a team playing in the Motor City and on October 5, 1932 the club was renamed the Red Wings."

Coincidence or not, the new name and logo marked a turnaround in the team's fortunes. The Detroit Red Wings made the playoffs in their first season, and won their first Stanley Cup in 1936.