I think that the Sorting Hat is unique, and that Thinking Cap is, as another poster said, in reference to "put on your thinking cap". It may have been (at least in the context) a title the Sorting Hat gave itself. Of course, many writers (especially in poetry) capitalize things for emphasis, not proper noun-ship.
So you think maybe it just highlights the fact that this is a hat that actually thinks!
so that the reader is more introduced to the type of magic that is available in the wizarding world? I guess that fits quite well. But still, it really stood out to me as more significant than that.
I'm certain that while there may or may not have been other hats like the sorting hat, there were variations - Ravenclaw's diadem actually helped you think better, Fred and George enchanted some hats I believe, and I'm sure I recall that there may be something similar that students are banned from using during exams...
This fits with more of what I was thinking initially.