So, you also think that archers are just the same as light infantry and skirmishers, just because they could in theory fill that role?
Horse archers are just versatile, and they could easily take the traditional roles of scouts and skirmishers and could harass the enemy if it was needed. But their main role was to outflank and inflict heavy damage to slower enemy formations, something their number allowed them to do at range.
Mongol armies usually had something between 20k and 40k horse archers. Even if every 4th of them inflicted just one casuality, we're looking at casualities that exceed small to medium medieval armies. That's way beyond the scope of what supporting light units were meant to do.
They were primarily archers, but their horses gave them the advantage of high mobility, so not only were they able to negate the usual disadvantages of archer units, it also allowed them to fulfill other roles, such as scouts, skirmishers etc. But they were archers first and foremost, and the main unit of the mongol army.
The Mongols used Heavy Lancers as their melee shock cavalry, which is about the only role horse archers couldn't fill.
So no, they're not the same, not even close.