First, notation sometimes evolves, sometimes not. I don't think $P(x,y)$ is bad, but it is somewhat old, I believe, although still in use but not quite à la mode, somewhat like French these days. :) It's not as bad, say, as the notation $\sin^2 x$ or worse (for me), $\ln^2 x$, which I first learned meant $\ln \ln x$ but my students use for $(\ln x)^2$. But what I have to say should not be taken as promoting the old notation. As $P(x,y)$ is used less and less, fewer and fewer people will be familiar with it. That in itself is argument against it.
Let $E^2$ be a Euclidean plane. A coordinate system on $E^2$ is a bijection $E^2 \leftrightarrow {\bf R}^2$ (with certain properties). The point $P(x,y)$ is a notation represents a point $P \in E^2$ corresponding to the element $(x,y) \in {\bf R}^2$. The point $Q(-1,3)$ is similarly understood. When this convention is well-known, it is not particularly hard to understand.
There are a couple of competing impulses in current mathematics. One is that notation should clearly indicate a single object that depends only on the symbols out of which the notation is constructed. I can trace this back to Frege, but it seemed to take hold in the 20th century. The other is that notation should be efficient. The desire for efficiency quite often leads to a violation of the first principle. The notation $P(x,y)$ is certainly efficient, the parentheses being arguably extraneous; but since they are used to denote a coordinate pair, they might also be defended. The implicit coordinate mapping is rarely important. Usually when the notation $P(x,y)$ is used, it is in a simple geometric problem and there is only one coordinate mapping. The context can be made clear by words, which is often a better way, in my opinion, than with abstruse and superfluous notation.
When I was young and impatient, I did not like $P(x,y)$. The $P$ seemed extraneous. In many cases the letter $P$ was not needed for the problem, but the author seemed to feel every point needed a letter attached to it. Part of the problem was that I thought ${\bf R}^2$ was the same as the Euclidean plane. If they are considered distinct, and they often are in modern synthetic geometry, then $P = (x,y)$ is certainly wrong, unless $(\_,\_)$ denotes a function ${\bf R}^2 \rightarrow E^2$, which normally it does not. If the plane and ordered pairs are identified, as I was led to think in my early years, then the notation is okay.
The normal way is to define or posit a mapping $\phi \colon E^2 \rightarrow {\bf R}^2$ (by introducing coordinate axes), and let $P = \phi^{-1}(x,y)$. But this seems cumbersome. So it seems to me one can chose which is the lesser of two evils: the notation $P(x,y)$, which does not conform with current functional notation; and the notation $P = (x,y)$, which treats the plane as identical with ordered pairs of real numbers. The current trend in usage is for the latter, of course.
