The horse is a very interesting creature. They were once, in prehistoric times, tiny. The Arabic name for them translates as "Star dogs". Archaeologists and paleontologists are unsure where they evolved. Many a tradition declares their hooves cause the thunder or their breath causes the wind. They are sometimes associated with the ocean and its foam. They are large herbivores with physical power and intelligence.
In France there is a national park which contains a marshland. The surrounding lands are, like most European countries, heavily farmed and otherwise occupied by humanity. In this marshland there is a breed of small wild horse who, although born all different colours, mature into white coated adults. These are the wild horses of the Carmargue, along with their black oxen friends. Most faerytales and myths of unicorns originate from France. Could it be that the unicorn, in the past, found itself hunted and losing its habitat to farming, might have been cornered in the marshland where man had difficulty hunting or farming. And, if their numbers were down, could it be possible they bred in with the local wild horse. There is no reason for a horse to become white as it matures, most other animals turn white in winter for camouflage.
I suspect that the unicorn "horn" is actually a form of psychic focus arising from the horses third eye. If it was hair, nail, or bone we would have found evidence of it, unless it was ALL collected and used beyond recognition. That is the hypothesis any way.
The Roman description of a unicorn sounds remarkably like a rhinoceros, and probably was.
However, there are sources beyond the Empire, particularly the Gauls and France of a distinct white horse with a horn. It could just be that the white marsh horses were seen and there was a reason to believe it had a horn, or someone misinterpreted someone else, who may have been giving an example of spirituality or a trick of the eye.