Depending on one's point of view that might be thought of as the "real" source of the paradox, although paradoxical groups are clearly the more interesting mathematical phenomenon in this whole story

Though you still need the axiom of choice in the process of lifting the paradoxical decomposition from the free group to construct the subsets in the usual statement of the Banach-Tarski paradox

Derivations if you think of them as directional derivatives, equivalence classes of curves if you like to view them as the infinitesimal movement of a particle. For me, vectors in a chart make the most sense in the context of submanifolds to visualize the tangent space as a concrete subspace.