The alicorn entry of the Oxford English Dictionary shows this in its etymology section:

< Italian alicorno unicorn (14th cent.), medicinal substance made from the horn of a unicorn (16th cent.), ultimately showing a variant or alteration of classical Latin ūnicornis unicorn n. (compare discussion of Romance forms at unicorn n.). Compare post-classical Latin alicornus (1551 or earlier).

The same dictionary shows these as the word’s two main meanings:

1. A (mythical) animal supposedly resembling a bull with wavy horns, sometimes used as a heraldic device.

2. A horn regarded as or alleged to be obtained from the unicorn. Also: the substance of this. Cf. unicorn’s horn n.

In recent times, the word alicorn began to be used to refer to the winged unicorn itself, and apparently it was reanalyzed as a compound word derived from the Latin words ala, “wing,” and cornu, “horn.”  According to the usual procedures for creating nominal compounds in Latin, the Latin compound adjective *alicornis may be formed from these two words, and it may have the meaning “having wings and a horn.”  It is analogous to the Latin compound adjective unicornis, “having a single horn,” from unus, “one” or “single,” and cornu.  This unicornis is used substantively to refer to the mythical unicorn.  The substantive use of *alicornis should refer to the winged unicorn: if an unicornis (unicorn) is a horse having a single horn, then an *alicornis (alicorn) is a horse with wings and a horn.

In the compound *alicornis, the word ala appears as the combining form ali- (as in the compound words alifer, “bearing wings,” and alipes, “with wings on the feet”), and the cornu is given the Latin adjectival termination -is (which replaces the final u of cornu).  The English versions of unicornis and *alicornis, unicorn and alicorn, disregard that adjectival termination.