Yeah, all of their backstory I only knew from anime, magazines, and game's manual book.
Nothing from the game (not counting game gear version since it's basically from anime).
Outside from the anime I only know these (from magazine and manual book) :
1. I remember reading in magazine that deep inside Sarah hates Jacky for being overprotective.
That's why she's kinda easy to be brainwashed into fighting him (compared to Vanessa) since the root is already there.
That's rather different in anime. They're super close in the anime.
2. Akira is defeated by Kage-Maru in fourth tournament and having a breakdown for a while.
3. Jeffry is searching for his rival (Devil Shark) that has been kidnapped by J6.
4. Pai realizes that Lau didn't have long to live and finally respect his desire to pursue ultimate strength.
5. Eileen kinda had a crush on Pai. (If this game was on Switch, I'd definitely put the game in
this thread). She enters the tournament in order to get to close to her.
6. Lei Fei's goal is to defeeat Lau to see the secret of Ko'en-ken.
7. Aoi is the type of a girl who get frustated easily every time she loses. The only way to get rid of her frustation is by entering tournament and fight again.
8. Goh is ordered by J6 to kill someone in tournament but his plan is ruined by Dural's appearance and hates Dural since then.
9. Kage is just like what you said. To save the soul of his mother.
10. Lion enter the tournament to learn about his father's involvement with J6.
11. Brad is a playboy but her preference is strong women. That's his motivation to enter the tournament.
12. Shun Di is searching for his student that has been kidnapped by J6.
13. Lau is ready to face the last fight of his life.
14. El Blaze is feeling jealous in Wolf's victories in the heavyweight division. He wanted to surpass him.
15. Wolf saw a small mark on Dural's body that was very similar to symbol he had seen in his nightmares. He wanted to know what it was.
16. Vanessa volunteered herself to protect Sarah from J6 but then got back to Jacky's house unconscious and some memory of hers was missing. Probably J6 tried to brainwwashed her but failed and she escaped. She enters the fifth tournament in order to get a clue what happened.
17. Jacky wanted to destroy J6 so that Sarah could live a normal life. After what happened to Vanessa, he enters the fifth tournament to help her too.
The reason of fighting for each character indeed exists. SEGA is just too lazy to put in the game.