OK, this is a concrete claim. How would you address the sorts of styles of play which are typically addressed by the signature mechanics of FATE? (I would take these to be character aspects primarily, and how they are tied into the mechanics, I know this varies to some extent between incarnations of FATE-based games).
For my part I see nothing in D&D which provides anything like the dynamics of compelling an aspect in FATE, or the mechanics of scene framing which it expects. 'classic' D&D is entirely bereft of anything here. I will note certain exceptions which help to prove this assertion:
- D&D in general has classes, which are pretty good at defining characters, but they are very generic and offer no mechanics or framework for leveraging them. At best DMs might do things like take account of a character's class in determining how NPCs interact with them (IE offering an army command to a fighter, and an advisor role to a wizard).
- Fighters specifically have 'domain mechanics' at high level which DO verge on something close to an 'aspect' in a sense. However, they are generic to all fighters and don't provide any mechanism to compel, nor any direct tie into structuring the plot. At best they act as guides to DMs and players, at worst as a simple resource system (a class feature).
- Ability scores, particularly in early D&D, could be seen as largely something akin to aspects. Again, there's little to tie them to plot, though they COULD govern character's options in play (but there are few formal rules for this, only scattered subsystems like BBLG).
- Paladins have specific restrictions. These are pretty strong, and enforced with a big stick, so they do produce some results, but not in a way very similar to FATE!
- Alignment, depending on how the DM interprets it, might perform some of this work.
- Race could be sort of an aspect, but this is similar to the case with class, very generic and lacking any sort of ties to mechanics that would shape the plot.
Specific editions sometimes have other little tidbits. 2e's XP rules are a bit of a way for the DM to compel certain things from the PLAYERS, but nothing works the other way at all. 5e has the optional Inspiration rule and associated 'character traits', but they are really kind of just tacked-on to the existing game paradigm.
I think that this is missing the mark. The question isn't "Can system X ape the mechanics of system Y" but rather "can system X provide the same playgoal/outcome as system Y". The idea is to look at whether a given system can evoke a desired feel, not whether it does that evocation the same way. For instance, no one plays Fate to evoke the Fate's mechanics. They play Fate because it lets them evoke the kind of experience they players want. Instead of evaluating whether or not D&D can do the mechanical end of what aspects in Fate do, we should be looking to see what aspects actually do, and then if D&D can do that.
My answer is that aspects evoke meaningful character choices in game by using those character choices to allow success or add complication to play.
That said, D&D can do this. I do this regularly in my D&D 5e game by taking what players have given me as important about their characters in backgrounds and traits and letting those improve or hinder action declarations in play. Frex, I had a player that had caravaner as a background, who's story was that she grew up on the trail with her father's caravan, trading around the realm. For this, whenever a question of where a given settlement or landmark was or the best way to get there from here, her character knew, and had advantage on navigation checks between settlements. In Fate terms, she had an aspect along the lines of "I always know the way" or somesuch.
Now, if you want to talk about which system does this better, then Fate does, hands down. It's built around this kind of evocation as the centerpiece of play. No arguments. But I don't think comparing whether or not D&D has aspect-like mechanics really gets to the question asked. I think you can mostly do most play goals in most systems, but they're better served in systems that place those play goals front and center rather than systems that pay lip service to them or mostly ignore them. D&D does what it does, and I think it's a broad platform that can accommodate a wide range of play goals, but none super duper well. It's generalist and somewhat malleable. Other games, especially ones that focus on a theme or playgoal (or both) will do that better than D&D, oftentimes much better, but it's going to be hard to not find at least some space for that goal in the broad tent of D&D.
Now, all that said, D&D does fantasy. It doesn't do non-fantasy settings well. Other games (like Fate) are very much setting agnostic, so that's a strong point in their favor. If my goal is 'I want to play cyberpunk' D&D is of no help whatsoever, but I could play that in Fate.