Is this a feature in the main game? Because I was doing it for a while, but since have removed a couple mods, and can't seem to do it anymore. I'm not sure if it was included in a mod (only horse mod I've ever used is horse armour, which I've always used) or if it's in the main game.

I was doing it and can't seem to anymore. Is it a bug that's causing it to work/not work?

Umm...I don't think it is in the main game, nope, pretty sure it isn't.
The only mod that I know of that let's you do that on horseback (and talk to people) is Convenient Horses...

Edit: I never used the horse armor mod so I don't know what features it has.

You can activate things while on horseback in vanilla. At least you can talk to people, but you need to point directly at the object, which is very difficult to do in third person on horse. Most of the time you will be dismounting your horse.

Maybe that's it then, I'm just not accurate all the time. But I know for a fact it does work, I was thinking to myself how come I've never seen this feature mentioned before. Discovered it by chance, picked a butterfly while on my horse.