I can't watch the video where I am now, but I can comment on the phrases in question.
Both "the woman's card" and "the woman card" are technically correct, but they mean different things.
In the first, the possessive indicates that we're referring to an object (a card) that belongs to a woman. We may use it in the following way:
The company's full name and address was written on the woman's card.
In this case, we're talking about a physical card, most probably a piece of cardstock or a business card, which belongs to a particular woman.
The second is a bit of a colloquialism or idiom, which follows the pattern "the x card", where x is a noun. It is used in the set phrase: "to play the x card." The most usual x used in this phrase is race, but I've seen gender and age used as well.
Here is an example usage:
I was getting nowhere with my complaint, so I decided to play the race card.
Alternatively:
I wasn't going to make my flight, so I decided to find a sympathetic-looking TSA agent and play the woman card.
In both of these scenarios, the speaker is deciding to emphasize a particular trait -- in one case race, and in the other case gender. In the first example, the speaker is trying to make a complaint, but nobody is listening to him or her. They decide that they're going to try to make the claim that this is happening due to their race. They may say something like: "You're not taking my complaint seriously. This is because I'm black, isn't it?"
The second example is similar. The speaker is trying to catch her flight and determines that she's probably not going to make it to the gate on time. She decides to find someone sympathetic, and to play up her gender traits, which usually ends up playing on stereotypes of women being weaker. As a woman, I've only done this once and not in this particular context, but it involved finding an older gentleman in a position that could assist me, and then a bit a crocodile tears and hand-wringing.
This set phrase is a metaphor for playing cards. In a card game, you may play a particular card to win a hand or a trick. For example, I may "play the ace of hearts." (In this example, the word "card" is understood because "ace of hearts" refers to a particular playing card.)
The metaphor here is that you're playing a "card" representing a particular trait that will win an argument or cause the situation to go in your favor.
This pattern does not hold true for other uses of the possessive. For example, "children's toys" and "men's clothing" do not have idioms "children toy" or "men clothing."
Similarly, the most common use of this idiom, "the race card," doesn't usually have a possessive equivalent "the race's card." (I'm sure we could come up with a convoluted example, but such a thing would not be in common usage, and its meaning would come from its use in context.)
Making the claim that one phrase is more popular than the other is irrelevant in this case, because they mean two very different things.
In case it matters, this answer is referring to common usage in American English. There may be different usages or meanings in other dialects.
