Why do we call the English horn by its French name Cor Anglais and the French horn by its English name, why does this exist? What does the French Horn go by in France? Cor Francais?
The English horn has no roots whatsoever in England or the English language. If the name were truthful, perhaps the Silesian Horn or Polish Horn would have been more accurate.
Historically, the English horn was associated with angels because of some common depictions in art at the time, and this led to the horn being called engellisches Horn in German.
"Engellisches" means "angelic".
"Englisches" means "English", and is pronounced nearly the same.
The resulting cor anglaise is most likely a transliteration that incorrectly took "Engellisches" to mean "English" instead of "angelic".
So, even though it seems ironic that the English refer to their own horn using another language's name, it is a bit more understandable once you realize that the name actually existed in French before it did in English.
I am aware that this only gives some background information without actually answering the question. This is simply a case of a word being literally borrowed from a different language instead of being translated, transliterated or receiving a new English word altogether. Other examples of the English language borrowing words from French are rendez-vous and chef d'œuvre, but there are many more. If you want an accurate answer to why specific words get borrowed, I'd post in the English SO -- the users there probably know more about this than us musicians.
They are, in fact, two completely different instruments. 'French horn' belonging to the brass family, operated by piston or rotary valves, and the 'cor Anglais' being a double reeded instrument, thus part of woodwind.
The term 'horn' originated from the hunting horn, and developed with graet changes into, amongst other things, the French horn. Officially it should be called merely 'horn', although that confuses it with any other of the horns, like trumpet, trombone, et al. The French horn became morphed into the German horn, mthere was also an Austrian version.The valves were added to allow it to play more than just harmonics, and the hand in the bell gives it even more control over the notes it can produce.
The cor Anglais is a different beast altogether, part of the oboe family. Other woodwinds - clarinet, sax, have a single reed, whereas oboe and cor Anglais has a double reed. Neither a horn nor possibly English. It's thought that the 'horn' part came from its resemblance of those played by angels in Middle Age paintings.In Middle High German, the word engellisch meant angelic as well as English, so there's potential confusion there. However, I can't discover why 'English horn' was then translated into the French language to give us 'cor Anglais'.
Ironically, it was used in parts of Europe from around 1740, under the Italian 'corno inglese', didn't come into popular usage in France till 1800, and took till 1830 to be used regularly in England!