This is one of those words (like Klenex) that is actually a brand reference but has been around (and used) so much that it is the standard name for everything of the type. To be more accurate, what most people call a frisbee is in general a "disc". The "Frisbee" is made by the company Wham-o... and is not known among Ultimate enthusiasts as a very good disc... it is better than nothing any day, but not good for games.
"You want to use a Frisbee? Man...go get the good disc."