Sure, you can exercise choice over the small region of your body consisting of reproductive organs, but the rank and file Democrat doesn't want you choosing what to eat or drink, where to educate your child, whether or not to belong to a union, and what type of light bulb to use in your home.
It might seem illogical or inconsistent, but it really isn't. You'll notice that the lady with the big black bow and Obama sequin shirt says (at 0:54), "I think a lot of people, uh, make harmful decisions. There's alcohol, there's cigarette smoking, and so I understand the government trying to intervene because the American taxpayer pays a lot of money for fat people."
Clarifying, isn't it? Abortion, not harmful. Eating donuts, pretty much the worst thing imaginable for both your fellow taxpayer and the planet.
Then we come to the portly lady with the decked out cowboy hat who says (at 1:46) that people should basically only be allowed to "make the choices that have good results," thus taking sequin-shirt's philosophy of allowing people to make any choice so long as it does no harm a giant step further. And, by her internal logic, abortion is not only not harmful, it's a "good result." She later disagrees that people should have the right to choose when it comes to other sorts of activities, but sums up her position as "pro-choice on everything about everything about everything. A person should have a right to choose, but it should be an informed choice and government has a role in that."
That summary —as incoherent and inconsistent as it might seem—perfectly encapsulates the liberal understanding of "rights." To a liberal, Rights aren't conferred by God or nature, they aren't inherent, and they don't derive from the Constitution or some sacred text. Instead, rights are doled out by the Democratic Party. And you don't have the right to disagree.