The chance is 0%
Let us begin with this observation: with 13 pair of cards in play, that accounts for every card in the half-deck. There are no cards remaining out of play.
So let's construct a scenario in which the game would never stop. To make that happen, each player must have one high card (say, above a 7) and one low card (7 or below). Since each player has a high card and a low card, and passes the low card, nobody will wind up with a pair, since all the low cards are moving at the same time, just chasing each other around the table, and never matching the (above a 7) high cards that aren't moving.
Sounds good, right?
The problem is that this precise situation can't actually occur; there are only 12 cards in the half-deck above a 7 (pairs of 8s, 9s, 10s, jacks, queens, and kings), but 13 players. Which means that we're guaranteed to have at least one player whose highest card is a 7 or lower.
Contrariwise, there are only 12 cards in the half-deck that are below a 7 (pairs of aces, 2s, 3s, 4s, 5s, and 6s). This means that we're guaranteed to have at least one player whose lowest card is a 7 or higher.
So over time, in this game, it's the lowest cards that will be circulating around the table: the pairs of aces through sixes, plus one seven. And since we're guaranteed that at least one player's highest card will be a 7 or lower which will remain in their hand as other cards are passed around the table, that player will eventually form a match, ending the game.