Is cardinality the only property of a set? Just a basic set, as in collection of distinct elements. Not a mathematician, so I'm not sure.

No. Because of limited comprehension, sets can only be identified via properties as subsets of another set, so for example:
$A = \{1,2,3\}$ and $B = \{2,4,6\}$ are both subsets of the superset $\Bbb N$, but the second set is a subset of the set of even natural numbers (yet another set), and the first does not have this property.
However, as sets (with no other structure upon them but "set-ness") they are isomorphic, which is a relaxing of equality to a more general "equivalence". This is sometimes useful when we don't care about the particular nature of the elements (which larger set they came from) but only "how big" our sets are.
