What does $X=x|Y=y$ actually mean by itself? I know that:
$P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}$
But I'd like to understand what $X=x|Y=y$ means by itself, for example $(X=x,Y=y)$ means $X=x$ and $Y=y$. Would $X=x|Y=y$ mean? that $X=x$ when $Y=y$?
A example to make it easier (take these as observations):
$X=\{1,6,7,8\}$
$Y=\{0,1,19,5\}$
$(X=x|Y=y) = ?$

This can get quite philosophical fast. But, Judea Pearl's book  Causal Inference for Statistics Section 1.3.3  provides a nice intuition, the operator $|$ implies a filtering of the data in the frequentist interpretation. An intuitive example would be two variables having bounds $P(X>a|Y<b)$, so conditioning implies filtering the data, i.e., removing parts of the data where condition $Y<b$ doesn't hold first.
Regarding if $X|Y$ is an event or not. It isn't an event in plain form but the resulting filtering operation leads to an event. Philosophical part of this then, what would be the Bayesian interpretation and even conditional probability can exist in isolation.
