I have noticed that every follower I get will default to using a hunting bow at range, instead of the bows I give them. I tried pickpocketing them but they don't even HAVE one. Is it just a graphic glitch or are they just refusing to use the bows I give them? I noticed if I don't give them a melee weapon,t hey equip my bow but that is the only way.
I'm pretty sure this is a bug. I've been having inconsistencies with my follower and bows too.
It's seems that they completely ignore Daedric Bows altogether. As for all the other bows, I've found it really depends on the arrows that you give them and vice versa. I've found if you give them an arrow that is a "grade" higher than the bow, they will default to the hunting bow and use the better arrows. This can go the other way around too. For example if you give them a dwarven bow and glass arrows, they'll take out their hunting bow and use the glass arrows. If you give them a glass bow and drawven arrows, they'll use the glass bow with iron arrows.
I've only found this true with Argis, and to be honest, I don't look at it with a microscope, so I may not be totally right. This is just from casual observation.