This is pretty much a translation thing, but suddenly I've started wondering why the gym badge given out in Saffron City is the Marsh Badge.
It does not fit neither the currently active Psychic-type gym, nor the former Fighting-type gym.
It's likely the names for the Marsh Badge and Soul Badge got mixed up during development and it just stuck. The could have fixed it in the anime and Yellow version like how Ghost was made super effective against Psychic, but I guess maybe they considered it such a little detail that overall affected nothing they decided to keep it.
So now we gotta come up with an excuse why the Psychic Gym has the Marsh Badge (and I also guess why the Poison-type Ninja Gym has the Soul Badge). Let's see:
Psychic Gym/Marsh Badge: A "marsh" is a wetland that dominate plantlife is herbaceous plants, plants that have no woody stems. It's what makes it different from a "swamp" which dominate plantlife is trees and a "mire" which is an accumulation of decaying plantlife. Because of this, a Marsh could be considered a more serene environment where the land and water blend together in a peaceful balance. In addition herbaceous plants, better known as herbs, are used for medicinal and therapeutic purposes (both scientific & folk remedies), and are also common ingredients for magic spells & rituals, all you can easily link back to either Psychic powers or the mind.
Being awarded the Marsh Badge shows you have a tranquil mind, a balance of strategic thinking and adapting to the situation. You also understand what each of your Pokemon has to offer, using their power and special skills to be the most effective. And particularly you understand what it takes to battle and use the mental and supernatural Psychic-type.
Poison Ninja Gym/Soul Badge: The "soul" is what "you" actually are and is what is controlling and being contained in the physical body. Your thoughts, feelings, memories, personality; all are components of the soul. While complicated, the soul can also be a fragile thing, easily swayed by sin & corruption; poisons of the mind. One can train their soul to be resistant to such temptations, and such training is what those practicing ninjutsu wish to master to become a skilled warrior. And while a ninja's training gives them resistant to corruption, it also teaches them how to use it as a weapon in their arsenal. For you see a ninja doesn't reject corruption, they understand evil is as part of the soul as good, but rather they want to be the master of their own destiny and be the onces to "choose their poison" then rather have it chosen for them.
Being awarded the Soul Badge shows you are in full control of your own soul, be it pure of wicked it was of your own choosing and no one else's. And you use this control to willingly manipulate others, most notably your Pokemon's, in becoming efficient weapons against those who would stand in your way. You are aware of the Poison-types affect on one's being, but also see the use of such power.
In-World Non-Pretentious Answer: The Pokemon League handles the production of Gym Badges and when making Sabrina's and/or Koga's they got the labeling mixed up. The paperwork to fix this small mistake is, of course, an inch thick and overly complicated that neither Sabrina nor Koga want to waste their time trying to straighten out a mess that isn't their fault so called it a wash. They then came up with my excuses above to make best of an annoying situation.
