Clarifying Probability Mass Function (PMF)
I believe they mean the following: "We will use $P$ to denote the concept of a probability mass function (pmf). $P(x)$ is the pmf of random variable $x$, while $P(y)$ is the pmf of random variable $y$. The two variables are usually the results of different random experiments, hence we use two separate letters, $x$ and $y$. As such, they will probably have different pmfs. But somethimes, it might happen that they have the same."

I think the essential source of confusion is that $P$ is not used to denote any single pmf, but is the name used to refer to a pmf. As a parallel, height_of(Stefan) and height_of(Mark) may be different, or they may be the same. height_of is just the name of the thing we are measuring, not the thing itself.

P(x,y) is thus the pmf of the joint variable (x,y). P((x, y)) would be a more unambiguous notation, but I guess is omitted for brevity. If you need more clarification on joint variables let me know. I hope this helps.