I was wondering if it was possible to extend the counting measure to give the cardinality of an infinite set as opposed to just ``infinity''. This extended function should satisfy non-negativity and take 0 on the empty set. I think that countable additivity should hold as, for disjoint $X_i$:
$$ \mu\left(\bigcup_{i\in \mathbb{N}} X_i\right) = \text{card}\left(\bigcup_{i\in \mathbb{N}} X_i\right) = \sum_{i \in \mathbb{N}}\text{card}(X_i) = \sum_{i \in \mathbb{N}}\mu(X_i).$$
I am not completely confident on cardinal arithmetic and just wanted to see if the above reasoning makes sense.
Sure, this would "work". To be more precise, given a set $X$, defining "counting measure" on subsets of $X$ in this way gives a map from $\mathcal{P}(X)$ to the class of cardinal numbers (or the set of cardinal numbers $\leq|X|$) which is "countably additive" with respect to the usual notion of addition of cardinal numbers. (In fact, it is additive on arbitrary disjoint unions, not just countable ones.)
However, this isn't very useful from the perspective of measure theory. The whole point of measure theory is being able to measure sets with real numbers, because real numbers have a very rich algebraic and analytic structure (which, for instance, you can use to define integrals of functions). We can't do most of this with infinite cardinals (for instance, we can't subtract them), so usually there's not any point in trying to be more specific than just defining certain sets to have measure "$\infty$".