It varies, both in terms of which game I'm playing, and what the particular group has agreed upon.
In one sense, class
absolutely means something concrete in my Dungeon World games. There really is no other Paladin but mine--that's been made abundantly clear. And, to a certain extent, there is no other Fighter but our group's Fighter (there are mercs, but only one "Equalizer"

), and
almost no Thieves but our Thief. (The Thief has a former love interest, gained as a follower at one point, who has essentially become the second part of a deadly dynamic duo with the Thief, their skills similar but complimentary.)
The case is strongest for my Paladin (who has almost single-handedly reinvigorated religious practice on their continent), and weakest for Wizards (who, as a whole, have become a fantastically important aspect of our campaign, what with their demigod-like Avatars and their Towers and their magic both subtle and world-bending). Particularly because we had a Kobold Wizard at one point, who was specifically unusual for being the only Arcane caster in the world (that we knew of, anyway) that
wasn't trained by the Conclave (the loose and fractious 'government' of the five Towers of Magic).
But in other games I like (4e) or have some interest in (13A), class really doesn't mean much of anything. The games explicitly and intentionally encourage players to re-interpret things as they see fit--refluffing, or even rewriting, where appropriate, to generate the right look and feel. A "bow Fighter" is a carefully-fluffed Ranger; an offense-minded Jedi is an Avenger; a prince could be either a Bard or a Warlord while still being exactly the same person; etc. The precise mechanics are only important in as much as they help you feel like a contributor and give you enough engagement, and you can assign to them (nearly) whatever fluff you like.
For me, personally? I'm sort of on the fence. On the one hand, I really like the idea that "being a Paladin" means something, signifies a particular kind of character archetype (self-sacrifice, service to a higher calling, etc.), just as "being a Warlock" or slightly further afield "being a Dragonborn." On the other, I hate it when I have a particular conception in mind, and the game I'm playing actively gets in my way when I try to implement it, even though it's neither exploitative nor particularly "out there" compared to the rest of the game. So I guess I would say I like it when the
default fluff is really cool and distinct, but the game makes it clear that the default fluff need not be the
only fluff and that player-generated ideas are just as good. So you can have "Monk" really
mean something, and you can also have "I'm not a Sorcerer, I'm a Firebender, I'm way more like a Monk than those dumb rampaging magic-flingers!"