I have been using my companion, Lydia, as a storage space until I can save up enough to get a house. However, when I try to retrieve certain armor from Lydia that she is NOT wearing and was not hers originally, the game says "can't take teammates' starting armor". Even if it was actually hers, she doesn't need all 7 fur hides. Why is this happening and how can I get it back?
You don't NEED to kill her. The other option is to pickpocket your stuff back. To make it easier to pickpocket, gain the perk poisoned, pickpocket enough of her stuff to reverse pickpocket a paralysis potion, and pickpocket her while she gets back up. This will cause a 100% chance of pickpocketing all of her items even if there's a 0% chance of getting the item. This may be the only option left to get your stuff back without killing her.
This happened probably because you told her to pick the items from the ground instead of giving to her in the traditional way (dialog --> I want to trade some stuff).
This method bypass the weight check (which is probably the reason you've used it) but also might bypass more important checks, so it might cause the item to become "default" - seen that in other answer here some time ago, can't find it though.
I fear the only solution is using console commands aka cheating.. just kill her, loot her corpse to retrieve all items then use the answers here to revive the poor woman.
Yeah this is annoying, but ShadowWizard is correct - using an NPC to pick up apparel such as armour because, say, they've reached their carry limit but you really really really need that pair of dwarven boots causes them to assume the armour is theirs.
Lydia done this to me on one save, I made a load of dragonbone armour and dropped it because I couldn't be bothered to walk back home from the forge, and when I tried taking it from her, the game claimed it was hers. This being on the 360 version and my autosaves already being used up, I had to gut her and take the armour back.
Whatever, I didn't like Lydia anyway.