In the international rules (also used in United States tournaments) the ways to draw that are usually a bit unknown or modified by non tournament players are:
1. Three move repetition - the same position appears on the board 3 times (doesn't have to be consecutively).
2. 50 move rule. If no piece has been captured or pawn has been moved, you can claim a draw after 50 consecutive moves. (This is the closest to what you seem to be describing).
There are no special rules for when one player only has a king left.
Usually players will have heard a variation of the 50 move rule. I've seen 5 moves, 15, 20... and now I've heard 13 Yes, this is an example of house rules. I woudln't expect a master to teach you this as if it were part of the standard rules.
Maybe it's used for a scholastic club to make sure games fit in the allotted time or so that new players are encouraged to know basic mating patterns (instead of repeatedly checking without making progress). Something like this I'm guessing.
Has anyone ever heard if your king is the only piece left that your opp has 13 moves to put you in mate or the game is drawn? I've always been taught that this is a rule, chess.com staff tells me that this rule doesn't exist, but i've been told and taught by teachers and masters who play this way. I trust the staff to be correct, however i cannot be the only one to know this. They tell me this is an example of "house rules"...