Shamir's Secret Sharing Scheme - prime security?
The prime can be just any prime, so it doesn't have to be chosen randomly (but of course it can). And the prime is made public, so it is not part of the shares. Public values are known by definition by everyone, therefore are not part of the shares.

But you're right, the quote is wrong about the size of the share, because it neglects $x$. However, it's really close, because $y$ is usually the same length as $p$ (uniform distributed), but $x$ can be chosen to be small.