I would be looking at games with some form of motivation mechanics built in - where the characters get bonuses or improvements based on their motivations and emotional states. Here's some games you might consider:
Primetime Adventures
PTA mechanics are very light, it does not have any tactical element, but it is very good at drama. It can emulate basically any story genre and would work well for pretty much any anime where you are not concerned with the actual game play being about tactical choices. Characters get "spotlight episodes" where they are more effective at dealing with problems than the other characters, which creates a natural narrative flow.
Blade of the Iron Throne/Riddle of Steel
If you do decide you want some tactical play along with your motivation mechanics, these two games do provide that. Although they're designed for Sword & Sorcery/Grim Fantasy, the basic rules of Spiritual Attributes (relationships, goals, ideals) are pretty easy to port, and much of the combat rules can be simplified if you aren't looking for hardcore swordfighting rules.
When you pursue your Passions/Spiritual Attributes, you get extra dice to succeed. When you want to improve your character, you spend down those numbers, so you have periods where you are very awesome, riding high on those pools, and periods where you are relatively weak, but you've improved your actual character stats a step, and will have to build it back up via roleplaying.
Hero Wars/Heroquest
Originally Herowars, then Heroquest, and now Heroquest 2, the core mechanics have been pretty close through the editions. Your character can have traits like "Strong 15" as much as "Brave 15" or "Doubting my own worth 13" and so on, and all of these traits can affect the character's success or failure. It can scale infinitely high, so if you wanted to do a game where you really focused on people powering up over time (or, getting more motivated) it works as well.
The only drawback is that all of these games tend to be somewhat rambly in the text, which can make the rules seem more complicated than what they are.
The Shadow of Yesterday
The Shadow of Yesterday is a fantasy game, but it is very easy to hack to any genre you want. Characters are powered by "Ability Pools" which drain down as you spend out of them, and are replenished primarily by interacting with other characters ("Spend some time talking about how you feel with another character, replenish your Instincts" etc.)
You gain XP by following Keys - motivations. Key of Romance, Key of Comradery, Key of The Vow, etc. These are also easy to hack to any setting as well.
Tenra Bansho Zero
The rules are moderately bound to the setting, so there'd definitely be some parts you'd have to consider which rules you'd use or how, but the core aspect that would be very valuable is the Fate-Aiki-Karma rules.
It basically works like this:
1) Roleplay in a way that's entertaining - get Aiki awarded from the other players.
2) Roleplay towards your relationships/emotions (Fates) - and that Aiki turns to Kiai -extra dice, basically action/hero points.
3) Spend out those Kiai for bonsuses, and after you've spent an amount, you will need to change or let go of your Fates - your relationships, ideals, and outlooks will need to change, you will have to grow as a character.
TBZ is very anime, so you can probably port some good amount of the rules with a bit of work.
FATE
FATE games, like Heroquest, let you take any character trait and make it something game-able and useable for advantage in play. "Loyalty to my friends" can be a way to get a bonus. When your Aspects work against you, you get FATE points, when you want to use them to get bonuses, then you spend FATE points.
FATE games are pretty popular right now as the genreless system for a lot of folks.
