The concept is from Chinese. In Chinese, 風 was principally "wind", but wind (and by extension changes in temperature) was also believed to be the source of various aliments to the physical body.
The Japanese word kaze originally only meant "wind". The sense "(sickness) cold" was influenced by Chinese. Note though that it originally was not limited to the cold but also included other sicknesses as well. The word 風邪 was typically read as fūja and simply meant "bad wind". It was not until a little more than a hundred years ago that these kanji were used express kaze "cold". (It is ateji.)
中風 and 痛風 are medical aliments borrowed from Chinese. Note that 中風 is rather literal in meaning: 風に中る (kaze ni ataru), to meet with (be struck by) the (bad) wind.