Characters in Sorrow in Haven have two die types set by their class - Endurance and Damage.The Endurance Die
This is pretty common - different character classes have different amounts of hp. Endurance in SiH is a bit different. It is like hit points, but only represents stamina and stress. Vitality (another stat) actually represents the amount of physical damage one can take before getting killed. Endurance is also a resource that characters can spend to increase their chances of success.
At 1st level a character gets a roll of their END die to determine stating END (15+END die roll), but after that every character gets 5 END when they level up. Early on it became obvious that characters with a big END die were just more capable then those with smaller END die - at everything (because you can spend END). So an initial bump, but long term END is fairly even across the classes. Adventurers are adventurers.
So why still have it? Because the END die represents how effective the characters are at recovering END when they rest. Fighters recover from activity (which is usually physical in nature) faster than wizards. Vitality recovery still sucks for everyone, as intended. So if HP aren't the measure of combat prowess any more (and they definitely aren't in my game), it has to be ....The Damage Die
I never liked classic D&D where characters couldn't use certain weapons. I understand why (game balance reasons) - a fighter is awesome at fighting so they get the best weapons. This was somewhat mitigated over time with proficiencies and whatnot, but it is still hella clunky. I also totally hate the idea that a dagger is less dangerous than a sword in the hands of a capable opponent. Hence, class-based damage. Fighters do 1d8 when they attack, wizards 1d4. Why? Fighters know how to fuck enemies up with anything in their hands, wizards not so much (they get spells, so you know, compensate in other ways).
This also allowed weapons to not be an endless list of stats, but rather descriptive items items that further the character idea. All "medium close combat" weapons are identical for stat purposes, just describe it however you want. Mace? Fine. longsword? Fine. Chain covered in salt-crusted thorns? Sweet. Gear, at least at a basic level, is then just an extension of the vision a player has for their character.
I'm not the first (by a long shot) to use or even suggest class-based damage. I did it decades ago with the BX D&D game I ran as well. It works. It is fun. Arguments against it are boring.
"Game" Balance / Multiclassing
Both a character END die and Damage die can be improved. For all classes. If there isn't a specific advance within the class itself, there is always the option of the alternate "Broaden Your Horizons" advances. They have a level requirement and cause your XP totals required to advance to increase. Why? because the character is deviating from the archetype.
I hate "multiclassing". If a fighter 6, warlock 2, rogue 1 ... fucking Christ! You aren't anything coherent at that point. Your character is a collection of abilities and not an in-game concept. But having some options isn't unreasonable. A Dr. of Physics can also be awesome at singing, so why can't a wizard also deal some heft sword damage? The "broaden your horizons" advances don't change your class, your archetype, your focus, they just open some possibilities. Besides, there are tons of alternate powers and advances that anyone can choose.
Some "From Play" Examples
One of my players has gone all-in with a concept of a gun-slinger. He is playing a fighter that has deviated and chosen the alternate power "Black Powder Enthusiast". This guy is fast and deadly when it comes to shooting people with guns. Like seemingly way over-powered at 2nd level. However, the moment the tide turns and melee is upon him, things don't go as well. The character is so hyper focused on shooting pistols that anything else is mechanically wanting.
In the previous campaign, we had a wizard that was all about causing damage. Massive damage. Destructive spells, physical damage, etc. In addition to being a war mage (increased damage die on spell effects) he also advnaced his general damage die. He was a fucking destruction machine, but when it came to interacting with people he was awful (and this was played oh so very well).
Are these flaws? Nope. I say they are strengths. You can craft a character that excels in one thing at the cost of other things. And it is obvious from the get-go. He knew in the last session that once those gang thugs closed in on him, things were about to go fucking sour, and they did. That wizard was shit at everything except blasting monsters apart and the player really leaded into it. The whole things was awesome.
I don't have one. I like making END growth fairly flat as it makes every combat dangerous, even at higher levels. I like class-based damage because it opens up character concepts without gimping them with abstract game rules. I like character classes as archetypes, hate multi-classing, but love customization options. Done rambling.
I hope everyone is having a happy (or at least reasonable) holiday season. Stay Safe. Trump & his cronies & supporters are pieces of shit. Black Lives Matter. RPGs are awesome. Keep playing and doing awesome stuff. Don't be gate-keeping assholes. Orcs can just be orcs. Free Healthcare for Americans. Fuck you, 2020.