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How to explain the fact that unicorns only let virgins ride them?
Why does the unicorn even care if the rider is a virgin or not?
And why do bicorns do exactly the opposite? Bicorns won't let any virgin ride them.

Those two creatures externally resemble horses with horns on the forehead. They are not necessarily normal horses or related to horses but they are still restricted by biological limits, no magic involved.

This is a misconception that has arisen from a mix of superstition, folklore and pseudo-science. Also some back-and-forth between languages.

In English there are distinct words for virgin and virgo. In some lating languages, though, there is just one word for both. It was said that unicorns would only let virgos ride them; when the tale made it into English speaking countries the whole thing was mistranslated.

Why virgos? Astrology has it that each zodiacal sign comed with some personality traits. Those born under virgo are neat, orderly and trim.

You may not believe in astrology, but the people who came up with the myth did. And as it turns out, unicorns are very virgo-ish. They will not walk on mud, and they will not allow themselves to be ridden by people who they perceive as dirty, such as being covered in soot or dirt.

So... In ancient times, when people would bathe once a year and most people believed in astrology, people would say it would take a virgo to ride a unicorn. When the myth reached UK, people mistranslated it, and now people think it takes an intact hymen to ride them 'corns.

Bicorns, on the other hand, like to roll on the mud. They also cover their legs in feces for thermal regulation. For astrologers, no virgo would ever ride them. For people with common sense, no one would ride them unless they are ok with needing a few baths later. Again, the myth got a whole new spin in English due to a mistranslation.

P.s.: this is not the original version of the myth, it's just something I came up with for this question.

1. Unicorns (I've edited to include bicorns)

This is actually a myth but a pervasive one and partially based on fact

In the old days there were no horse-whisperers - horses were 'broken' to the saddle.

The wild unicorn simply would not submit to being broken - it would behave like a bucking bronco in a rodeo. From exhaustion it might temporarily submit (but stand still and refuse to move forward). On recovering its strength it would be completely intractable again.

Horse breakers were predominantly men. The unicorn detects their maleness by its exquisite sense of smell. It can detect testosterone and indeed the slightest trace of semen.

In general married women found little or no success in training them, firstly because they tended to use the same methods as men - namely 'breaking'. Also because of the likely scent of semen however faint. (Hygiene standards were low in those days so people didn't tend to wash their private parts).

Virgins had neither large amounts of testosterone nor traces of semen. They also tended to be small because in those days marriage could take place at puberty.

Young girls are known to have an affinity for horses (these days manifested in My Little Pony) and are likely to tame by kindness and affection. One such little girl discovered her ability by accident when she encountered a unicorn in the forest. She felt no fear and had no concept of breaking a horse. She was able to interact with and even ride the unicorn. She ran back to tell her friends and parents and the myth arose from there. Thereafter virgins were used in any dealings with unicorns that required them to be tamed or even ridden.

EDIT
In response to a comment:

"...in the old days there were no separate bedrooms. If there was a bed, it was for everybody in family. So I would hardly say that virgins had no male odor on them. – L.Dutch"

My reply

It could be that unicorns mainly frequented the royal forests because they were protected from poaching there. The only virgins that encountered them would be daughters of nobles and would likely have a separate bedchamber and handmaid if only to obey the dictum, "Children should be seen and not heard". Note: Even royal hygiene was hit and miss at that time however pre-puberty there is little B.O. of any kind.

P.S. This raises the question of whether virgin boys would qualify. Maybe they would but I suspect that they would be brought up in a male tradition of hunting or subduing forest creatures rather than being kind to them.

2. Bicorns

The Bicorn is a creature (often described as a part-panther, part-cow creature with a human-like face[1]) that has the reputation of
devouring kind-hearted and devoted husbands
https://en.wikipedia.org/wiki/Bicorn_and_Chichevache

The obvious answer is that bicorns refuse to let almost anyone ride them. All they are interested in is devouring kind-hearted husbands. Maybe they beguile men to ride them and then after a suitable interview to determine whether they are kind-hearted or not, stab them with their horns in a backward movement or throw them off accordingly.

Sensing virginity is an innate ability that members of the unicorn family have

Unicorns, like many animals, mate for life. They invest a tremendous amount of effort raising their young, and their courtship rituals aren't just a wonder to behold, they also occupy a significant portion of each unicorn's lifespan.

Unicorns have a few magical abilities, but in terms of them breeding there are only three that matter: their impressive glamour display which they use to woo potential partners, their ability to sense whether a unicorn is their own offspring, and their ability to tell whether or not a potential partner is a virgin.

Due to male unicorn's unfortunate habit of killing their rivals' offspring, a tendency to avoid non-virgin (except for their partner) unicorns was selected for.

A side effect is that unicorns also avoid humans who are not virgins.

Bicorns have similar abilities, but live in radically different conditions. Due to the hazardous terrain, partners will often die young, so the willingness to switch partners is more advantageous. In addition, circulating among the bicorns is a strain of Equine Magical-Deficiency Virus, which causes them to appear as if they are still virgin. Because of this, bicorns have the opposite tendency to unicorns.