ASH
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Ash's rivals in Ring Racers are stand-ins for
gods or godlike beings from the Fire Emblem universe.

Flicky is the stand-in for Yune, the goddess of chaos who takes on
the physical form of a small bird while traveling with a lore-relevant
human from the Fire Emblem: Radiant Dawn game.

Heavy Magician is the stand-in for Bramimond, the legendary dark mage
from Fire Emblem Blazing Sword, who morphs his face and voice
to match that of whoever speaks with them.

Rouge is the stand-in for Embla, the goddess of closure that can transform
into a giant bat divine beast, and is the main antagonist from
Ash's storybook in the Fire Emblem Heroes mobile game.

SCREAM TAIL 8BIT
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The rivals are stand-ins for other Paradox Pokemon
from the Pokemon Scarlet/Violet games.

Tikal is the stand-in for Flutter Mane: she forfeited her body
to seal her (and Chaos's) spirit away inside the Master Emerald 
to stop Chaos's rampage. This happened long before 
Sonic was even born.
She's basically a ghost from a long time ago, just like Flutter Mane.

Pulseman is the stand-in for Miraidon, as they both have the ability
to control electricity.

Jack Frost is the stand-in for Iron Bundle, they both have
the ability to utilize ice for attacks during combat.

ROBOEGG
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Bark and Bean are characters that, including Roboegg, debuted in the 
Sonic the Fighters game.

Metal Sonic rivaling Roboegg is a nod to Wusstunes' Youtube video
that recapped a tournament to decide which of Roboegg or Metal Sonic
is the more overpowered Sonic the Fighters...fighter. I won't spoil
that result here.

Sorry, Honey. ¯\_(ツ)_/¯