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.
Not really
The skills are:
Strength
Athletics
The echo does not need (nor can it) climb or jump or swim, etc. It simply is moved 30 feet at a time by the knight. No check is required for this.
The echo is not granted the ability to lift or carry or drag objects so no check is able to be made.
When grappling or shoving, the rules say that it is the knight that is making the attack/grapple/shove check, not the echo.
Dexterity
Acrobatics
As above, the echo does not need to attempt to maintain balance or swing from ropes, etc. It simply is moved 30 feet at a time by the knight. No check is required for this. In addition, it is immune to all conditions so it would never need to make an acrobatics check to escape a grapple.
Sleight of Hand
The echo is not granted the ability to pick pocket (a la Arcane Trickster Rogue's ability Mage Hand Legerdemain) or manipulate the environment in any way.
Stealth
The echo, as above, is simply moved by the knight. It does not possess the ability to move of its own accord and so it can't move stealthily or attempt to hide. Moreover, since the knight is the creature making the attack, the rules for unseen attackers, etc apply with regard to the knight, not to the echo. This can get weird, depending on how you interpret the rules (this closed question is relevant).
Intelligence
The knight would be the one to make any of these checks because the knight has the knowledge and ability to reason, not the echo: Arcana, History, Investigation, Nature, Religion.
Wisdom
Animal Handling
The knight would be the one to make any such check because the echo can't manipulate the environment.
Insight
The knight would be the one to make any such check because the echo can't reason.
Medicine
The knight would be the one to make any such check because the echo can't reason and the knowledge would reside with the knight.
Perception
The knight would be the one to make any such check, even after it gets the feature Echo Avatar at level 7 because the rules clearly say "You [the knight] can see through your echo's eyes and hear through its ears."
Survival
The knight would be the one to make any such check. The echo can't track or reason, doesn't need food/water, etc.
Charisma
Deception
The echo does not possess the ability to speak. It is possible to imagine a situation where the echo is used as a decoy of some sort (call it a passive deception). But as the echo is created by (and an image of) the knight, the onus would be on the knight to change its appearance to look like whatever it wanted the decoy to look like and therefore, as a DM, I would still require the check from the knight, not the image.
Intimidation
Similar to Deception but with the added caveat that, if a knight were to, say, try and intimidate a creature by making that creature think that it was outnumbered by more than one knight, I might rule that the echo's presence helped in some way but it would still be the knight that makes the check.
Performance
Similar to Intimidation or Deception.
Persuasion
Similar to all the other Charisma-based checks.
Given the limited capabilities the echo has (which is to say none that the knight doesn't take itself) I can't think of a situation where the echo would be required to or capable of making an ability check on its own.
Best Answer
The Echo Knight requires a lot of work.
I've played an Echo Knight, and then later DM'd for an Echo Knight. My experience with the subclass has been that the DM is going to be making a lot of rulings. As a player, my DM and I had to start making a list of interaction rulings for the various features. Knowing this, I did the same thing when I DM'd for an Echo Knight. So the short answer here that doesn't get into the calculus of the rules is:
The DM and player should discuss and agree upon a ruling and then just be consistent moving forward.
How I ruled it.
I'm not going to hem and haw my way through the technicalities of trying to interpret this question from a "rules as written" perspective, because honestly, the rules as written here are woefully unclear. Instead, I'll just tell you how I played the class, and how I ruled it later as a DM.
To me, the ruling that seems the most probable as the intent of the feature is that when using Echo Avatar, these two things are true:
With this scheme, if you make an attack with your corporeal body, for the purposes of resolving the attack, you are blinded, which means you have disadvantage on the attack roll. Ruling it this way is for me motivated by something from my days as an amusement park manager. I used to sit in the office and watch security cameras, one of which was aimed at myself. I would try to throw balls of paper into the trash can or at my coworkers by judging my target's position on the camera screen, rather than actually looking at it. As you probably guessed, I was much less accurate this way than if I was actually looking at it, and that is how I look at trying to attack corporeally while looking through the echo's eyes. To me, the primary advantage of this ruling is that it is simple and easy to apply. In my recollection, there were no real ambiguities to resolve for this ruling's interaction with other combat rules. You know how the rules work when a creature is blinded and deafened, and you know how the rules work for a creature that isn't, and that is all you need to know to make rulings about this feature when you go with the ruling I outlined above. This ruling did not create the need for further rulings. Sometimes when you come up with a ruling, further play leads to needing more rulings as your ruling interacts with more features. That didn't happen here.
It should be noted, however, this is still better than being blind without the echo avatar. The rules for "Unseen Attackers and Targets" state:
If you are just blind and deafened, you are guessing where your target is. With an echo avatar surveying the field while you are swinging your sword blindly, you don't have to guess your targets location. "I can see that there is a skeleton standing right in front of me! I will swing my sword in that direction."