The bottom line is we don't exactly know the final word on what it was. But let us not let our modern day version of a mystic horse with wings that glows next to children sidetrack Gods meaning in all of this is my contention.
In Numbers 23:22 KJV says unicorn yes, but we need to study why to understand it better as well. I see TinTin has done a good background check on this. Other versions of the bible such as NIV, NASB, Even the Catholic Dovay-Rhiems bible historically call this KJV version of a unicorn other terms. Such as wild Ox, and Rhinoceros. Now, to note, even the Scholars tussle over this translation, but we know God didn't. The original Hebrew word "re'em" is the foundation to this. A urus or auroch, as TinTin has already mentioned was an ancestral wild version to the domesticated cattle of today. Re'em flourished in early historical times. They were a dangerous, very strong creature a lot like the Asian buffaloes, yet they are now extinct.
When the first Greek translation of the bible was prepared around 250 B.C. the re'em were already rare. Hence, why the Greeks had no word for it. So, they used the word "monokeros" which means one-horned. When it was translated into Latin they used the word, "unicornis" which means one-horned. But does not mean a horse with a single horn.
Now this is where it get's interesting...in 1611, rather than speculate, the KJV just made up a new english word "unicorn" rather as they did with "baptismo", Greek for full-water immersion, rather than offend King James I, who had been sprinkled, they invented a new word, "baptism" which could mean "sprinkled, poured, or immersed."
My point is regardless of what it is, we need to guard our hearts from mysticism or anything that drives us to a corner of thought outside of God's intentions. It is clear to me it is a literal animal for it speaks of the unicorn in context to other known species; Dogs, eagles, calves, lions, lambs, peacocks, bullocks, goats, horses, donkeys, Job 39:9-12.
In Job 39:9-12 it points out a description of this unicorn "whose strength is great" but it is useless for agricultural work, refusing to serve man or "harrow [plow] the valley." Now that simply sounds more like a Rhino than a horse with a horn to me, but whatever!
I ask the question too, where is the dodo bird today? ...It isn't! Yet , we don't doubt it once existed right?!!. We need to see the enemies angle in this too. Our modern connection to both angelic like connections and mystic endeavors, like that of the rainbow today as well, or anything that sets us apart from Gods point in the passages mentioned. I pray we guard our hearts from the subtle stepping aside from Gods point to allow for outside influences to delight in versions we want to see over what is to be seen by way of His Spirit.. Anything outside of faith is a problem, even subtilely so.