How to type a Simplified Chinese degree symbol on a regular English keyboard?
Chinese word processors have the option to use U+2103 ℃ or U+2104 ℉, to get over the typographical problem of displaying a degree much closer to the C or F than otherwise allowable in fullwidth text. 

However, it is also true that the Chinese 度 is used more commonly in everyday text and in most newspapers; my personal contact with ℃ has been in academic papers.

Both of these are in the Unicode block "Letterlike Symbols", alongside ℏ, the  "Planck constant Over Two Pi" U+210F and Ω the Ohm sign U+2126; most common is likely the ™ trademark sign U+2122, also repurposed in modern Chinese's Internet slang.