Is this improvised card game is a known game?
Since players choose their next card it is a variant of Colonel Blotto game (unlike in War where players just pick the top card in their deck).


  In Colonel Blotto, players are tasked to simultaneously distribute limited resources over several battlefields. The player devoting the most resources to a battlefield wins that battlefield.  The player that won the most battlefields wins the game.


In your variant, players contest over 13 battlefields. Each side poses 1+2+...+13=91 resources (armies), however they must assign the resources to battlefields in set amounts (the value of cards).