They can just say tell us what is copyrighted and we'll take it down but you might as well have a warehouse of a few hundred thousand items with say a third of the items that belong to another company. Is it reasonable to say just tell us what is your's when you removed the items without their permission. It's obvious in this case what belongs to the wronged party but you are saying you'll only return the items the other party identifies.
If I have a company and a third of the items belong to another company illegally. The other company will contact the authorities who will then ask "Which items are yours?" so yes, they have to identify thier items. You obviously have never been robbed before. Same thing happens in a robbery. The police have your items but you must go to the police station and identify them before you can have them back. It is your property. It is your responsibility to keep it safe. If something happens to it, it's your responsibility to fix it.