The word Chanukiah was not used until fairly recently. It was invented by Hemda Ben Yehuda towards the end of the 19th century. So all of you who say 'menorah' is not the correct word in English, or any other language, for the "chanukiah," what did people call it before she invented the word "chanukiah?" Probably - menorah. (I saw on wikipedia that it's called Hanukah Lamp in Yiddish.) Maybe if there was some confusion they would add the word Hanukah to distinguish it from the Menorah in the Beit HaMikdash. But that probably only occured among Torah scholars when discussing ... The Hanukah menorah is a memorial for the Menorah in the Beit Hamikdash. That is where the tahor oil was lit after the war with the Greeks and the recovery of the Beit Hamikdash. Menorah can be translated as lamp.
The word chanukiya does not appear in my (rather old) Merriam-Webster Unabridged Dictionary, the Merriam-Webster online dictionary, my Random House Dictionary, or my Oxford English Dictionary. However, menorah is listed in all four sources, with each source giving at least one of the possible definitions as a Chanukah lamp. (The OED does this by giving a usage example.) Yes, I know that מנורה means something else in Hebrew, but the Chanukah lamp is most commonly called a menorah in English. I am sorry that some people are uncomfortable about this meaning of the English word menorah in relation to the meaning of its Hebrew cognate, but when asked to translate something into English, we should do so. Phrases like משלוח מנות do need to be transcribed into Roman letters because they have no reasonable English equivalent. But the word חנוכייה does have a direct English equivalent, and that equivalent word is menorah.
The HEBREW word מנורה refers to a lamp, and particularly to the one in the Temple. The ENGLISH word "menorah" means a Hannukah lamp, with a secondary meaning of Temple lamp. The word that Anglophones use for something is, by definition, the correct translation of that thing into English!
Well, that simply raises the question of what "correct" means in regard to a language. In a language that recognizes a central authority, such as Metropolitan French with the Académie Française, one can say it is correct, if accepted by that authority. English has no such authority, so of it is accepted by a large number if people for communication in English, it is correct, at least in that group.
That is a tricky issue, isn't it. I would certainly accept that it is incorrect in some particular branch of Judaism, but as long as I still hear lots of American Jews referring to it as that, I would say it is correct in American English. This is not uncommon, of course. I would, for instance, compare it to the use of the honorific "reverend." In my denomination of Christianity, it is only ever used with another title (Reverend Mr, Reverend Dr, Reverend Fr), and I was certainly raised to think of referring to someone as "Reverend Jones," for instance, absolutely incorrect. I do, however, hear many other Americans use the title that way, so I would have to say it is not incorrect in US English.
If you are really interested, I would suggest looking at what the speakers of Yiddish called it two centuries ago (or far less likely speakers of Ladino three centuries ago). That would have been what Jewish immigrants to the United States called it. Since the number of US Jews was vastly more than that of UK ones, I would suspect that their terms became accepted in Jewish English around the world. It's also the reason why I think of the holidays, greetings, etc. in their Yiddish pronunciations.