As best as I can tell, here's how the game decides which character attacks for a given tile:
[note: different star levels of the character complicates tie-breakers (more below.)]
1) Highest damage for that color
2) Highest level character
3) Position on character selection screen (1 Center, 2 Left, 3 Right)
How many covers each character has in that color's power does not matter.
Here's my evidence:
A) I had a level 17 Bag-Man who does 12 damage in purple, alongside level 83 Storm and Thor who only do 9 damage in purple, and the purple tiles belonged to Bag-Man.
B) I had level 83, 2-star Storm with 5 blue covers and level 82, 2-star Black Widow with 5 blue covers, both of whom do 43 damage per blue tile. But blue tiles belonged to the higher level storm, no matter the character selection order.
C) I leveled up Black Widow to 83, she and Storm now tie on stars, level, blue covers, and blue damage, but now the blue tiles are assigned according to the character select screen (center out-ranks left and right, left outranks right.
D) I had level 41, 2-star, 3 green cover Ares and level 41, 2-star, 5 green cover Wolverine who both do 25 green damage. Green tiles followed the character selection order. So we know that number of covers doesn't matter, otherwise Wolverine would've won the tiebreaker both times.
The only one I can't explain is that I have a level 40, 1-star Black Widow who does 7 damage in red. I teamed her with 7 different level 40, 3-star characters (1 cover each, none red) who all did 7 red damage. Even when I put Black Widow in center, the red tiles always went to the 3-star characters EXCEPT for Daken (other characters: Blade, Kamala Khan, Rocket & Groot, Sam Wilson, Spider-Man, Squirrel Girl.) When I looked up the seven 3-star characters' damage dealt at level 166, they all did 13 red damage, except for Daken who did 12. So if characters have different star levels but do the same damage, the tie-breaker might be where that color ranks for the character (1-6, each character having three strong colors and three weak colors)?
Anyway, hope that helps.