Yes, you can ride all year in Seattle. You just need to dress properly, and you'll benefit from owning a bike with fenders for the winter (or put fenders on in the winter).
The other thing I'd add is that if you like to climb you'll find the terrain here much more varied than in Florida (at least based on my experience in Florida). You can ride in cities, along the coast, on islands, on rolling hills, or up in and around the mountains. I personally think the scenery makes up for the amount of climbing you have to do (or getting a bit wet in the winter).