But I do remember it was some kind of an adventure mode that dealt with players going to certain areas and fighting against other characters, and upgrading levels and status and something like that. If somebody remember that mode, I'd be glad to explain it for me, but what do you think?
That's the Weapon Master mode. You do indeed travel across a world map and fight other characters in scripted battles, each of which is a chapter in an overarching network of interconnected storylines. Certain fights have special conditions, too, e.g. "fight an opponent with regenerating health" or "the ground is explosive if you fall onto it". There's also dungeons that you can enter, wherein you make your way through a series of rooms on a grid map that slowly reveals itself as you travel, each room being a fight against in an opponent in that dungeon's themed stage, culminating in a battle with the "boss" that you can fight as soon as you've found them on the map.
It's called Weapon Master because you collect and buy alternate weapons for characters as you progress through the world map. You also earn experience as you fight, which levels up your character - since it's saved to your player profile rather than each character, you can swap characters at any time and maintain your accrued level. Level progression doesn't actually alter your game or characters in any remarkable way, it's just for bragging rights and to show how much you've played the mode.
I could see something like that working in Smash, definitely. It'd be a cool and interesting alternative to the usual scripted Adventure mode and would allow you to play all sorts of different opponents in various challenging situations, and you could collect and buy equipment, powers and custom moves as you play. The Master Hand / Crazy Hand theme could allow you to take different routes or create different battle conditions with tougher opponents and proportionately greater rewards.
Great catch, though! I like the way you think, good sir, and thanks for reminding me of what a swell game Soul Calibur is!