How do they quantify this claim? Do they have a control group to test it against?
No, of course not.
In order to make this claim that vaccines keep you from getting as sick, you'd need two different groups of the same age and health, with one group taking the vax and the other not. Then get them sick, and compare the results.
As it stands now, barring a vaxed person actually dying from COVID, they can effectively maintain this talking point. Even if someone comes within an inch of their life, they could say, "he/she would have died without the vax." It's essentially a no-lose for them unless massive numbers of vaxed begin dying. And this probably will happen.