If you ride only a motorcycle for 6 months and drop your car insurance, then drive your car for 6 months every year, you can save money. I did this from 1980 to 1984 while in college, but it was not fun with the motorcycle on cold or wet days, and my storage and passenger space was limited.
To keep both a car and a motorcycle and save money, you would need to have a long drive to work and a small, used motorcycle for it to save money. You also need to factor in the tire costs- my motorcycle needs new tires about every 8,000 miles, and they cost about $350 to buy and have installed by a mechanic. The bigger and faster the bike, the more tires cost and the faster they wear out. Some people with Hayabusas need a new back tire every 2 months for street use, less for track use.
Moneywise, an old 3-cylinder Geo Metro car that you can buy cheap will be the most economical for areas with a cold winter, but a guy at work that had one said they do not have much traction in the snow, because they are so light- it would be best to put snow tires on the front and carry 100 pounds of sand bags in the back for snowy weather. He drove his over 200,000 miles before the engine started to burn some oil.