Can you explain why "the only measure available is indeed the ordinary amplitude-squared measure"?

It's kind of an abstract mathematical fact. If you are fine with that, I recommend reading Everett's original paper, which includes this fact as equation 34.

According to the Wikipedia entry you linked to, a probability measure is a real-valued function, but X here is apparently just a number? What's the significance of your parenthetical note here?

The significane is that now we're talking about some "fundamental" property of the universe, rather than probability, which is more about our ignorance of what's going to happen. Um, so if that description of probability didn't make sense, the distinction won't make much sense.