A conditional probability $P(X |Y)$ where $X$ and $Y$ are two random variables can be represented in a graphical way as:
Now, my question is: does $P(X=x|Y)$ make sense?
If the answer is affirmative, this means that $P(X=x|Y)$ is only a function of $Y$, then $P(X=x|Y) = f(Y)$?
Please, clarify with a clear example.
"$P(X \mid Y)$ where $X$ and $Y$ are two random variables" makes no sense to me, so I will not discuss it.
But $P(X = x\mid Y)$ where $X$ and $Y$ are two random variables is a well-known notation (see https://en.wikipedia.org/wiki/Conditional_probability#Conditioning_on_a_random_variable). We have that $P(X = x\mid Y)$ is a random variable whose value depends on the random variable $Y.$ That is, when $Y$ takes on the value $y,$ then $P(X = x\mid Y)$ takes on the value $P(X = x\mid Y = y).$