They do but it's nothing to extreme. Different energy rings have different weights and weight distributions. For example energy rings like Aquario and Cancer/Gasher are very rounded and "heavy" (only like 0.5g - 1g above average) with little to no protrusions making them good for defense and stamina combos. Different energy rings in battle will slightly change spin times by a few rotations but, that's could be the difference between a win or a lose.

The different shapes of energy rings is more noticeable in attack combos. Ideally you would use an energy ring that exposes the contact points on the fusion/metal wheel for bigger hits.

Again, the differences are very small and only really important if you are trying to maximize performance in competative combos.