I understand that a combinatorial class $\mathcal{A}$ is a set of objects, with a function of size $|\cdot|_{\mathcal{A}}:\mathcal{A}\to \mathbb{N}$. With objects of size n: $\mathcal{A}_n=\{\alpha\in\mathcal{A}\;:\; |\alpha|_{\mathcal{A}}=n\}$, and a condition of finitude $\#\mathcal{A}_n<\infty$.

In a book "Introduction to Combinatorial Enummeration" by Dvid G. Wagner chapter 11, I came across the term "class of structures", and they define it as:

A class $\mathcal{A}$ associated with a finite set $X$ is another finite set $\mathcal{A}_X$, such that is $X$ and $Y$ are finite sets:

If $X\neq Y$ then $\mathcal{A}_X\cap \mathcal{A}_Y=\emptyset$.

If $X\neq Y$ with $|X|=|Y|$, then $|\mathcal{A}_X|=|\mathcal{A}_Y|$.

My question is whether these two definitions refer to the same thing. I am somewhat confused