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):
$X=\{1,6,7,8\}$
$Y=\{0,1,19,5\}$
$(X=x|Y=y) = ?$

The $|$ symbol in probability theory stands for “given”. You would most commonly see it used for conditional probability $P(Y|X)$, the probability of $X$ given $Y$. While it’s a slight abuse of notation, you could see something like
$$
Y|X \sim \mathcal{N}(\mu, \sigma)
$$
for $Y$ conditionally on $X$ following normal distribution. You would also see it to show some properties of distributions, like conditional expectations $E[Y|X]$, or variance $\operatorname{Var}(Y|X)$, etc.
Notice that something like $X|Y$ alone doesn’t make much sense. What would it be? “A random variable conditional on another random variable”? Conditioning is about the perspective you take when looking at the variable, not a property of the variable. You can “transform” conditional probability to joint, or marginal, or reverse it (Bayes theorem) with just simple mathematical manipulations on the distributions.
