> I am assuming you are aware that conditional expectation is a random variable
That's not what we're considering here, but things of the form X|Y=y for a concrete y. Even as E[X|Y=y], that's not a function, y is specified. Do you agree we shouldn't call X|Y=y a conditional random variable?
Oh absolutely for a specific y its not function (or a random variable) one usually thinks of Y as a variable and not a constant.
The expectation E[X|Y=y] is a fixed value. (Edit: it’s the expectation of the random variable “X|Y=y”, while E[X|Y] is a random variable because it’s a function of the random variable Y: for each element in the sample space there is a corresponding value of “y” and in turn there is a value of the expectation E[X|Y=y].)
X|Y=y (as used in the blog post being discussed) is a random variable: it’s a function from a subset of the original sample space (corresponding to the elements for which the value of the random variable Y is y) to real values (or whatever the image of the X random variable is).
More concretely its just another name for conditional expectation. I am assuming you are aware that conditional expectation is a random variable obtained via conditioning (equivalently as a piecewise approximation in L_2). If you arent familiar with that view point that would be the place to start. Kolmogorov, Neveu, Dudely, Billingsley will all cover that view point.