A mounted opponent who fails the save regards all creatures as enemies, including the mount they're riding
The enemies abound spell description states:
You reach into the mind of one creature you can see and force it to make an Intelligence saving throw. A creature automatically succeeds if it is immune to being frightened. On a failed save, the target loses the ability to distinguish friend from foe, regarding all creatures it can see as enemies until the spell ends.
The direct effect of the spell is clear: if the target fails its save, it regards all creatures it can see as enemies. Assuming the target is riding a creature (rather than a Broom of Flying or Carpet of Flying or something), and assuming they can see, they would indeed regard their mount as an enemy as well.
There are no special rules by default for attacking a hostile creature you're riding, or dismounting one
By default, the rules don't generally handle such a unique case; especially since the enemies abound spell was added in Xanathar's Guide to Everything rather than appearing in a core book, I'm guessing the designers didn't necessarily anticipate the spell's existence when writing the rules.
However, there is a potentially relevant optional rule on DMG p. 271, under the Action Options heading, titled "Climb onto a Bigger Creature":
As an alternative, a suitably large opponent can be treated as terrain for the purpose of jumping onto its back or clinging to a limb. After making any ability checks necessary to get into position and onto the larger creature, the smaller creature uses its action to make a Strength (Athletics) or Dexterity (Acrobatics) check contested by the target’s Dexterity (Acrobatics) check. If it wins the contest, the smaller creature successfully moves into the target creature’s space and clings to its body. While in the target’s space, the smaller creature moves with the target and has advantage on attack rolls against it.
The smaller creature can move around within the larger creature’s space, treating the space as difficult terrain. The larger creature’s ability to attack the smaller creature depends on the smaller creature’s location, and is left to your discretion. The larger creature can dislodge the smaller creature as an action—knocking it off, scraping it against a wall, or grabbing and throwing it—by making a Strength (Athletics) check contested by the smaller creature’s Strength (Athletics) or Dexterity (Acrobatics) check. The smaller creature chooses which ability to use.
The DM may choose to apply similar mechanics in the case of a creature affected by enemies abound that is mounted on another creature. Most mounts would likely not be able/intelligent enough to attack their rider, and thus would likely simply attempt to use their action to dislodge the rider if the DM decides to use these rules (assuming the rider attacks the mount first).