Could you play the games as different "classes" or something? Could you be an Enclave officer or a Raider or merc...whatever.

I hope it's understood what I'm asking.

No, but you could play as The Chosen One. How you played that character was up to you.

I understand that by saying playing 'roles' or 'classes' you mean whether you can play as a member of a faction?
In that the answer would be: not really.
You can join a few factions, but it won't have any major impact on the game (several factions with conflicting interests is pretty much a feature found exclusively in TES, the most you can expect from most other games is having to choose between 2 or 3 different paths during the story)

Nevertheless, you have a much larger variety of 'roles' and 'classes' (meaning: character builds, as there aren't really any predefined 'classes' as such) than in most other games.

Not in fallout 2. If I can remember, you were a tribal that was an ancestor to the vault dweller and it (the game) took place in the state of California. There were vaults that you could travel to and you could actually have a car to fast travel in. It was turn based, but you only played one character. In fallout tactics, brotherhood of steel, you were an initiate that worked toward becoming a Brotherhood paladin and the more missions you did and got farther in, you gained experience. You had a squad of I think 5 people and traveled in a Humvee. When one squad member died you could pick another from the list, ghoul, death claw, mutant and human that had different traits you could use. There were similarities from fallout tactics to this game. The ghouls in tactics worshiped an atomic bomb, there were brahmin and toward the end, you battled robots and the "collector". Fallout two had Harold the ghoul, and at that time just a small tree growing out of His head. They were fun games to play, but nothing compared to this game.

Harold and Brahmin have been in every Fallout game. Harold was missing from Tactics and Brotherhood of Steel.

Lets not forget Dogmeat. He's up there with Harold.

You could technically play various "roles" in FO1 and 2, however since your character had to handle all nonviolent interactions between your party (if you had one) and the NPCs there were certain skills you pretty much HAD to level up which put a limiter on what you could practically do. In addition, the various allies you could recruit were all specialized in some way but there were some skills none of them covered.

Interestingly, unlike FO3 you could get away with never leveling a combat skill since there were several NPC allies who could do that for you. Speech played a MUCH larger role and leveling it to at least 100 quickly tended to be a wise idea as otherwise a fair amount of content was blocked to that character. You actually needed about 70 or 80, but getting it to 100 or so helped when dealing with NPCs who inherently had a low disposition to the PC, and in the late stages you wanted it even higher for the Master (FO1) or the President (FO2, NOT the end boss unlike the Master) in order to have the advantage when dealing with them.

That's not really true, there have been quite a few games where joining factions play a part and in fact since IV TES has moved completely away from that as you can join all factions and the factions don't interact, so joining one does not exclude any others, they don't have conflicting interests to any real extent.
To the OP as others have said no not really, your first loyalty is to your vault/tribe.

Don't take this as an argument - I just can't remember playing any such game (with the exceptions of those were joining a faction is part of the story - such as the brotherhood of steel in the Fallouts, or those that do have optional (sometimes conflicting) factions but with only very few sidequests - such as the Reno families in FO2) so I'm asking out of mere curiosity... can you give me some examples of such games?

No problem these are just a few off the top of my head
Planescape had joinable factions based on different philosophies
The Gothic games
Drakensang
Wizards & Warriors
I think wizardry 8 had guilds as well
In fairness though TES 2 and 3 probably are pretty good at the faction aspect and are hard to top

Yes, I've played PS:T & Gothic3 (and I stopped drakensang because I couldn't stand the camera )
The big difference though, is that in TES (including Oblivion) the factions have extensive questlines that are not only irrelevant to the main quest, but they can even function as its substitute. Think of it like this: I am not likely to start a new game in PS:T to focus exclusively on playing through the Dustmen questline, whereas making a sneaky character in Oblivion for the sole purpose to complete all thieves guild's quests, while ignoring everything else, will still result in a complete and solid experience (in fact I think specifically Oblivion is better with such an approach)
In Gothic 3 the factions are so tightly related to the main quest that I would consider them more like 'branches' of it than 'independent' questlines.

But yes, those are fine... it's just that being on the official Bethesda forums I find it reasonable that when someone discusses factions will have in mind the much more sophisticated and unique approach of the TES - not that I think that every game needs such an approach mind you... PS:T & Gothic were fine the way they were and including extensive irrelevant faction questlines in them might have been a bad idea to begin with.