The wording in Seller Admin about "product-specific affiliate programs" is a bit misleading, as you actually have only one affiliate program for your entire seller account, which all your affiliates would join. Once they join your program, they can obtain their link code for your Common Hop Link (if you set one up) and/or any product-specific Hop Links you've set up. It is not possible to only offer certain links to certain affiliates; all of your affiliates can pick and choose among all the links you set up.
Regardless of which type of hop link a buyer follows, we set a referral cookie in the buyer's browser which expires in 6 months while we redirect them to the landing page URL for that link. Everything that buyer purchases from you during the lifespan of that cookie will earn the referring affiliate your specified commission -- the product-specific Share % for items that have one, and your Common Share % for all other items purchased.