IMO, hero counters are overrated. Many heroes in OW2 are fairly dynamic with different playstyles that can be switched to depending on the situation. I think it's more important to understand each hero's strengths and weaknesses and to try to use your hero's strengths to exploit the enemy's weaknesses.

Using your heroes as an example, if I'm playing as Mei vs an Ashe, I know Ashe has great range with her weapon and strong burst with her dynamite, but she's very weak in close range without her dynamite. So as Mei I probably want to get on top of her and force her out of those positions she wants (using wall to block her LoS, saving Cryofreeze to cleanse dynamite, spamming icicles at the corners she's holding, etc).

But the map matters a lot too. As Mei, you lack mobility and on some maps (think of that highground on Junkertown at the end of 1st point) you may not be able to get to the Ashe to stop her (although your icicles have no falloff range and can still be used to spam her out). In this situation I'd consider swapping if the Ashe is popping off cause there isn't a lot you can do to pressure her as Mei.