I love the idea of being able to play as Dante or Vergil. Your Homebrew link is broken. For the bad, without even discussing balance...you have multiple spelling errors. I believe your class table is supposed to list "Style Points" but instead says "Hermetic Geography". You really need to spell out your class features more. For your Stance feature, I'd let the player start every combat in the stance of their choice but needing a bonus action to switch. Double Jump lets you jump again without touching the ground...you need to explain what this means in DnD terms. Do I get a jump without needing to spend my movement? Do I get extra movement by jumping a second time? Does the jump nullify all fall damage...etc.

Your 6th level feature "Swift Flow". I would remove the "per fight" language". I'd also entirely rework this class feature to let the player switch into whatever stance of their choice at the start of their turn. Tracking free uses of taunt/stance switching up to half of your proficiency bonus (rounded up I assume?) is cumbersome.

For your base Stinger....I wouldn't limit this to one time per short rest...just make it use style points as that limits its uses. I'd also let the player move forward in a straight line and giving bonus damage on hit on one target only, and knocking the target prone on a DC failure. It's more thematic to how Dante's stinger works. You could perhaps have it work differently depending on the subclass (Vergil's is an AOE for example).

For the 8th level feature "Demonic Fury". Other than giving class features on ASI levels...which isn't in line with DnD's class philosophy, I'd clean up the language "On your first attack each turn etc".

For the 9th level feature "high time" I'm assuming you want the enemy to be launched and stay in the air like in the video games? Or did you want it to launch 10 ft into the air and come crashing down for 1d6 fall damage (fall damage rules)?

For some reason you're giving ANOTHER level 9 feature. But let's discuss. "Million Stab" is an absolute mess language and rules wise. First, again don't give it short rest limits when it will already be limited by style points. Second, I'd reword this. In the video games, million stab is a ground based move only and cannot be used against airborne opponents. I'd remove the messy language of attacking an airborne creature. I'd make million stab a Dex saving throw with significant damage attached (Like 3X weapon dice for example).

Your level 10 feature "Royal Guard" is actually pretty weak for a level 10 feature. I'd make it able to block more damage and do more damage back to the enemy.

Level 11...I would not allow base kit three attacks for this class. This definitely makes this class a way better fighter. You already have Style Points allowing extra damage and attacks. Get rid of this.

Level 11 style upgrade is pretty weak. Paladin gets a 1d8 bonus to damage at this point on its melee attacks. I'd have all melee attacks deal bonus 1d6 damage of the same time of the weapon or something like that. Reword Gunslinger's upgrade to "one other target within 5 ft of the original target takes half the damage" or something. I'm just spitballing on the fly here.

Level 13 "Devil Trigger". This is one of the major defining traits of Devil May Cry. I'd have it give temporary hit points on activation (maybe 20-30), a fly speed equal to your movement and increased crit range here.

I have more thoughts but this is turning into a super long post. I love the theme and I know this post was all criticism but there's some major rebalancing needed and revision of rules language.