In mathematics, you can use any notation you like as long as you clearly define the symbols you use and the resulting notation is unambiguous to a reader.
Having said that, it will generally be easier for both you and the reader if you fit in with common usage, but this is just for convenience and following common usage is not the same as symbols being "reserved".
People use many notations for the probability of an event $X$ other than $P(X)$, for example Pr$(X)$ is also very common and there are lots of other variants such as $\pi(X)$ or $p(X)$.
I would strongly discourage the use of $P(P)$.
While that notation is technically possible, because it is possible to keep the probability function $P()$ separate from the event $P$, it nevertheless is needlessly confusing.
If you do want to use $P$ to represent an event, then it would be wise to prepare for that by choosing a different notation other than $P()$ for the probability function.
Mathematical notation, like language in general, is a fluid thing.
The same author may use different notation for the same concept at different times, depending on the audience and the context and on what other notation needs to be used in the same document.
It's not necessary to do exactly what other people do, but it is necessary to be clear and unambiguous.