The primary goal of a set collection mechanic is to encourage a player to collect a set of items. For example, players collect and harvest different types of beans in Bohnanza, and they collect Monuments in Ra.

boardgamegeek.com

Set Collecting is one of the oldest game mechanics out there. It may very well be older than the idea of card games since ancient games may very well employ it. Rummy is the first game that comes to mind that uses Set Collecting. ‘Rummy’ “is a group of matching card games notable for similar gameplay based on matching cards of the same rank or sequence and same suit. The basic goal in any form of rummy is to build melds which consists of sets, three or four of a kind of the same rank; or runs, three or more cards in sequence, of the same suit.” Wikipedia. ‘Go Fish’ is another old game that employs the Set Collecting mechanic.

Some newer games that use the Set Collecting mechanic: Ticket to Ride, Settlers of Catan, Alhambra, Zooloretto.

Although there are many games that incorporate Set Collecting as the primary theme of the game, this is more often used as a non-thematic way to resolve play. For many games it just does not make sense to play Rummy to resolve the conflict within the game unless that fits into the game scenario.