Can I use Team Rocket's Ampharos' "Darkest Impulse" Ability to put 4 damage counters on both of my opponent's evolving Pokemon if they evolve two Pokemon from their hand during the same turn?

Yes, you can. The phrase "doesn't stack" means that if you had two Team Rocket's Ampharos in play, you would not be able to use both Darkest Impulse Abilities to put 8 damage counters on each of those evolving Pokemon.

Relevant cards

SV10_74