People carry all kinds of different cards. For example:
- a credit card
- a debit card
- an ID card for work or school
- a gym membership card
In some situations, you can just say "my card" instead of saying what kind of card you're talking about. People will usually be able to understand which card you mean. For example, if you're going to the gym with someone and say:
Oh shit, I forgot my card!
the listener will know that you are talking about your gym membership card.
In the example above, it's obvious that "my card" meant "my credit card" because the context is paying for a meal.