Like the Latin roots of the word unicorn, uni meaning "one" and cornu meaning "horn", there is a Greek word for the same beast, monoceres, whch shares some root words with the same meanings: mono meaning "one" and ceres meaning "horn". With this in mind, you would think the word Alicorn would be just as simple.
Wrong. Alicorn seems to be a bit more flexible with its root words. As you've already stated, it is the horn of a unicorn, but the root word of ali can be easily misconstrued. Check out this redit page and read all comments to get the drift: https://www.reddit.com/r/mylittlepony/comments/18aja5/adventures_in_etymology_alicorn/
Apparently ancient man had no idea what to make of the true animals of the world with horns, so we get a lot of different versions of the myth of the creature. The horn of a rhinoceros is still used in medicine today, which is another reason these magnificent creatures are being hunted to extinction. The Rhino's name comes from the Greek root words rhino which means "nose" and, once again, ceres, same as the Monoceres which is what it was considered once upon a time. The Orynx and many other animals with horns and antlers have also been subject to this, such as the narwhal which has been mentioned by someone else on this thread. It may even by that people found horned creatures like goats with a mutation of one horn on its forehead. There was a deer that was discovered recently that did have this mutation.
Now I did a search on the same subject a long time ago and came across the Ethiopian Unicorn, which is the earliest record of an African unicorn, however this creature has two horns, not one. This does not, by any means, mean it is not a unicorn in ancient times, but it may not be the same creature you’re seeking. The Greek term, Cerapter, which is a combination of the word ceres meaning "horn" and pteros meaning "wings", supposedly is the correct term, but barely anyone even knows this word so I cannot verify this with 100% certainty.
On another note, the term Pterippus could be the possible correct word for the winged horse. Pegasus is the name of the specific horse that sprang from the neck of the once human mortal she-monster Medusa, beheaded by Perseus and impregnated by Poseidon whose symbol is the horse. This is the horse that is captured, tamed, and ridden by Bellephron in the Greek myth.
I got that a good chunk of the information above from this article. Check it out for some even more interesting mythical horse creatures.
Just a footnote, but the unicorns of true myth were not horses with horns but a combination of the features of different animals, and they were nasty. They were not necessarily evil, but they were beasts to ancient man that embodied the pure unbridled fury of nature. They could kill a small army of men just as easily as the more aggressive dragons of myth could. They didn't like humans at all, but because virgin women were viewed as pure, it was considered the only thing that could tame them long enough for hunters to kill it and acquire its power. This is why most people today believe they are pure creatures that are for little girls only. I believe the only truly good unicorn was the Asian version which was used in court to judge who was guilty and who wasn't, but I challenge anyone out there to do their homework and prove me wrong.
There are many different types of unicorn myths in the world, and I'd like to encourage people not to stick with the very common European white unicorn. However, they don't have to be beasts like in myth. You can make your up your own. I prefer to think of them like the Asian dragons, wise and intelligent, a mix of both modern and original flavors.
Also, if I have anything incorrect, please doesn't hesitate to step in to correct me.