how to implement sense of cooperation in card games?

I believe bridge is a cooperative card game, Rummy with partners.

Look up the rules and see if that will help.

I don't really see this as any different than making a coop game in general.

Creating coop is as simple as creating a scenario that nobody can handle alone:
- Skill tests where all players need to contribute their cards in order to succeed (Like BSG)
For example: "You draw the repair the hole in your boat card which has a challenge of 7". Each player is only allowed to put one card in and the maximum skill is 5. During every situation, each player needs to contribute to the overall succeeds.

- Multiple challenges that happen at one time: (Ghost Stories, Ygdrassil, Space Hulk: Death Angel)
For Example: One player should be playing cards to "Unlock the Door" while the other player is "Holding off the Horde". Each player is working on a separate task that contributes to the overall "success".
No single player could handle all the issues that occur so they are forced to work together to survive.

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Also, it's helpful to reduce the number of "Help Yourself" cards and increase the "Target OTHER player". For example, allow the Medkit to heal other players instead of yourself or the Sniper Rifle to support OTHER attackers.
These mechanics will literally FORCE players to work together. These cards will make it clear even to new players that you should work as allies.

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Finally, DO NOT PUT IN COMPETITIVE FEATURES! If you want players to cooperate, they can't be punished for doing so. Telling a player that "You gain points for killing the monster!" or "You split XP with all heroes in a fight" leads to a situation where players are rewarded for acting alone.
Reward players for behaving the way you want them to play the game.

Maybe have a secret goal that each player has that differs from the rest but for everyone to achieve their goal they need to play cooperatively building towards their goal while the others are doing the same and whoever reveals that they have completed their goal that player is the winner?... For example I will use a zombie game, my secret goal is to kill 15 zombies attacking other players, I am helping the other players by attacking their zombies hence working cooperatively yet once I complete that goal I am the clear winner.. hope that helps, not sure it the idea will work into your mechanics

Maybe a building type game where each person plays a different role and that role determines how many VP they get at the end. They have to work together to achieve the end result although achieving their criteria provides them with more VP.

Are you talking a custom card game or a playing card game?