Sounds like you'd need to set up the player control to use an analog control system. I presume an easy analogy for what you want is the way that Smash has its movement? Slight pressure walks, full pressure runs?

I'm not too familiar with scripting Unity directional controls like that. Just hoping it points you in the right direction. This game, and most fighting games, use digital controls for the input. it's either Pressed or it's not. Some games have gotten around that by registering how long the direction is being held. So you could theoretically make a "move" that happens if the direction has been "charged" for a certain amount of frames? Just hypothetically thinking here.