E.g. Dungeons & Dragons has a SRD licensed under OGL (Open Game License), for the 3rd and 5th editions, but not all classes are in the SRD. Are the overall characteristics (not just the description texts) of the classes not part of the SRD protected by copyright to prevent reuse (even in non-OGL RPGs)?
Technically no, but be careful with what that means
You cannot copyright ideas. That means you cannot copyright the concept of small folk from a quiet village traveling deep into the wastelands of evil with an ancient artifact to confront a villain reborn, or an orbiting military academy where children are forced to compete in a brutal zero-gravity combat simulation, or a spellcaster whose magic comes from their bloodline.
Copyrights apply to expressions of creative ideas. That is, you don’t copyright the concept, you copyright the text, the images, the tables, the whatever-it-is that expresses your concept to others. So those concepts aren’t copyrighted, but The Lord of the Rings, Ender’s Game, and the Player’s Handbook are. The text of D&D is copyrighted, and none of those copyrights are due to expire any time soon. That’s why you need a license—like the OGL—from the copyright holder in order to use any of that text.
Moreover, copyrights include rights to derivative works—which means that, if your work is based on copyrighted work, even if you don’t actually use any of the copyrighted text, tables, images, etc., it still falls under their copyright.1
All of which means that, in casual speech, “copyright” is often used to extend beyond its strict, legal definition. We might speak of Frodo Baggins being copyrighted, when in reality the copyright applies to the text of The Lord of the Rings as well as to all derivations of that text—and any other text that uses the Frodo Baggins character will, in fact, be a derivation.² In fact, even if you changed the name, if a court decides the character is recognizably Frodo Baggins with the serial numbers filed off, that might still be considered a derivative work.
So can an RPG class be copyrighted? No—an RPG class is a concept, an idea. And class names are often real words, which no one can lay claim to, so you could certainly publish an RPG class named fighter, sorcerer, or wizard. But the Player’s Handbook text describing them is copyrighted. And that copyright extends to all derivations of that text—if you base your work on the D&D text and just change the words to synonyms, you are still derivative work.
On the other hand, “sorcerer” is a real word. The concept of magic derived from one’s bloodline goes back to antiquity. Wizards of the Coast probably couldn’t make a case just because you have a “sorcerer” class with a “bloodline” feature. But if you very obviously recreated the D&D sorcerer—down to specifics—in your new system, then they might have a case, even if you write new text and reference new mechanics. Even more so if you went and did this with every PHB class and subclass. But we can’t really know for sure; sometimes very obviously derivative things have gotten a pass.
For an example of this, consider The Sword of Shannara by Terry Brooks. It has very strong parallels to The Lord of the Rings. My description in the first paragraph could have applied to The Sword of Shannara as well as it did to The Lord of the Rings. As Wikipedia puts it, “Some accused Brooks of lifting the entire plot and many of his characters directly from The Lord of the Rings,” which I would agree with (and I liked the book well enough, and think it has some minor independent merit—as Wikipedia continues, “others praised its execution despite the lack of originality.”) But The Sword of Shannara was published, and the Tolkien Estate—famously litigious—didn’t sue. So there is some room here, though it’s notoriously difficult to find.
Ultimately, copyright law is subjective. We can’t know for sure how a court will decide any given case. And even if you would prevail in the end, defending a lawsuit from Wizards of the Coast would bankrupt most people. Your best defense is to use the OGL, but that comes with requirements. Your next-best defense is to make sure your RPG is very clearly unrelated to D&D.
Note: in this scenario, you have copyright too—you can’t publish your work because it is derivative of their copyright, but they can’t publish it either because your new derivation is something you have a copyright on. In this situation, you have to either come to an agreement wherein one of you allows the other to publish it despite the other’s copyright, or you publish it jointly somehow, or you simply agree to disagree and no one publishes it.
Some derivative works could fall under fair use, e.g. for parody, but there’s a lot that goes into fair use and basically none of it applies to creating RPG material.