Well, if you start the game in summer, your hideworking skill will be lower in summer during the winter, and hence you'd get a better outcome in the winter. The quality of the result of the skin making process is not dependent on the type of skin used, only on skill, input material quality, tools, and RNGs. The discussion concerns skin of the same quality (thus having an identical name), not various qualities.
Winter fur and summer fur are distinct in the game with the winter versions being more valuable due to having a thicker pelt, but it's currently not possible to distinguish between them apart from dividing them into like quality stacks and use trading to find out which stack is the most valuable one.