Amazon could easily solve this problem but they've decided it's more profitable to let it fester. Whether or not it's a good decision in the long run, the same way Sears permanently ruined their reputation by selling shit through their established in-house brands, remains to be seen.
They don't necessarily. Amazon combines items from different sellers together in their warehouses, and doesn't sort them out when picking orders to ship. So if I send them a high quality doohickey to sell with FBA, a customer who orders my doohickey might actually receive Mr. Cheapo's knock-off doohickey because as far as Amazon is concerned they're the same thing and got placed in the same bin in the warehouse.
Sellers can opt out of this but then Amazon charges them more.
The fact that Joe Q Public can make a marketplace account, ship a crate of counterfeit stuff, and have that be sold to buyers of “amazon.com” is absurd and explains the counterfeit problem.