I have some idea what would it mean, but I would like to have a precise definition. I've seen this term used here and there in computability theory, but I couldn't find a definition.
(My idea is that if a set A is in class C and there is B such that A/B is finite, then B is in C too.)
I'm pretty sure this means, for a collection $\mathcal C$ of sets, that for every $A,B\in\mathcal C$, $A\setminus B\in\mathcal C$.
You can use induction to extend this to say "for any finite sequence $\{A_i\in \mathcal C\mid 1\leq i\leq n\}$, $A_1\setminus(A_2\setminus(A_3\setminus\ldots\setminus(A_{n-1}\setminus A_n))\in\mathbb C$".
And also, for that matter,
$(\ldots((A_1\setminus A_2)\setminus A_3)\setminus\ldots)\setminus A_n\in\mathbb C$
I think that's what I think is meant by finite (not that $A\setminus B$ has finite size)
Actually here I have found another variation:
It says that whenever $A,B\in \mathcal C$ and $A\subseteq B$, then $B\setminus A\in \mathcal C$.
And just now I found one closer to your conjecture: there it means that the symmetric difference between the two sets is a finite set.
So it seems like it is used in different ways. This means you'll have to closely pay attention to the sources you care about and see what notion they're using.