No, it cannot (usually).
The class feature description of the echo knight's Manifest Echo gives no guidance here, but the echo knight subclass description is not all features - Explorer's Guide to Wildemount (p. 183) motivates for us exactly what an echo knight's Manifest Echo ability is doing:
A mysterious and feared frontline warrior of the Kryn Dynasty, the Echo Knight has mastered the art of using dunamis to summon the fading shades of unrealized timelines to aid them in battle. Surrounded by echoes of their own might, they charge into the fray as a cycling swarm of shadows and strikes.
But Thomas, that's just flavor text! Sometimes flavor text is not just flavor text, particularly when a natural application of the flavor text very clearly motivates how a particular ability functions.
In this case, I've bolded the key phrase; your manifested echoes are "fading shades of unrealized timelines". Your echo is an alternate timeline of yourself, which naturally means that the echo cannot do anything you cannot do, unless otherwise specified by the various subclass feature descriptions. If there is no alternate timeline in which I can move vertically or pass through walls, then there is no vertically moving or wall-pass-throughing echo to manifest.
But I have a flying speed!
Then your echo has a fly speed, maybe. If flying (read: moving vertically) is innate to your character, such as an Aarakocra echo knight, then alternate timelines where you move vertically are totally feasible, and your manifested echo could be one of them.
But what if I can cast fly? I'm going to call this DM purview, but I personally lean toward "Sorry, you cannot choose to manifest the echo that had just cast fly". Nothing in the feature descriptions allow an echo to generally benefit from buffs you have or have available to you.
But I can normally pass through walls!
Can you, though? Per PHB, p. 7:
an adventurer can’t normally pass through walls
I'm not aware of any racial or class features that just allow you to pass through walls willy-nilly. As for spells or magic items requiring activation, the reasoning I outlined concerning use of the spell fly applies.
As always, the DM may rule otherwise.
After receiving some feedback in chat, user GcL helpfully pointed out:
I think the presence of a floor in one timeline does not constrain the presence nor location of a floor in another timeline.
Following the logic of different timelines, the [echo] "in the air" is actually standing on a floor or ground that does exist in the timeline they're drawn from.
This is not an entirely unreasonable argument, so it would not be entirely unreasonable for a DM to permit the echo to move vertically or pass through walls, especially given the apparent ambiguity of the feature description.
