It’s not a tower, so it’s the other way around. At ground level, the cable only has to hold up the weight of a car. At a kilometer of height, it has to hold up that car plus a kilometer of tether.

At (about; adding cars to the tether changes things a bit) geostationary height (in this case I guess that could be called lunastationary), forces on the tether start going upwards, and tether size can go down as you go further up.

See https://en.wikipedia.org/wiki/Space_elevator#Cable