So invariants are generalized from geometry. Angles being invariant under isometries is probably the ur-example. If you have a triangle, the measure of the angle A is a property of the triangle. If you translate the triangle, the image of A has the same measure as A. Another definition of a property is a one-argument predicate; it's something you can say about an object. If you know an invariant holds before an operation, you know it holds after; that's what's interesting about them.

Something is only a catalyst in a particular reaction, so I would think it would only make sense to say "C is a catalysts" is a property of the reaction as a whole. My chemistry is beyond rusty though.