The echo is intangible
A plain English reading of "This echo is a magical, translucent, gray image" indicates that this is not a tangible phenomenon. In English, an "image" is intangible, therefore unless you add some adjective that necessitates it being tangible, it's intangible.
Your DM should make rulings
Since the rules don't explain what to do, you should rely on your DM. If a hostile giant tries to pick up the echo, it probably makes sense to play it as an attack roll and have the echo destroyed if it is picked up.
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.
Best Answer
Sometimes; it's complicated
The echo's movement in relation to other creatures is governed by a slightly obscure set of rules: The echo is not a creature, but it occupies its space, which is an attribute normally only ascribed to creatures. The rules which govern how space is occupied refer explicitly to creatures and movement, and do not map neatly on to non-creatures with no movement speed.
In this answer, I make three assertions about the echo's movement, each of which is less certain than the one preceding:
I'll try to justify these below, but in so doing I'll be making the best of a bad job: the rules for the echo's movement are poorly written, and require imaginative interpretation. I'll illustrate this by going through some of the factors which complicate the ruling.
Complicating factors
There are several factors which suggest that the echo's movement might not be inhibited by other creatures:
1. The Echo isn't a creature
The basic rules impose the following restrictions on player movement, which are assumed to extend to all creatures, except in the case of exceptions:
Since the echo isn't a creature, it's not immediately obvious that these rules have any bearing on it, or for other creatures wishing to move through its space.
2. The Echo doesn't have a movement speed
None of the rules for movement, including the above, can be applied directly to the echo, because the echo does not 'move' in the way that creatures move. It has no speed of any kind, and moves 'up to 30 feet in any direction' when mentally instructed to do so by the Echo Knight. This movement includes upwards movement and is more akin to a caster moving their Mage Hand than to a creature expending their movement to traverse space in a physical way.
3. The echo is an image
Even if the Echo weren't a creature and didn't have a speed, it might still be constrained by the common sense adjudication that corporeal bodies cannot pass through one another freely. The echo, however is:
One might argue that a 'magical image' can pass through pretty much anything. There's not a great deal of clarity here because the echo has an AC and no resistances/immunities. It's an image you can hit with a stick.
But there is one big restriction on the echo's movement
All of the above is overwritten by the ruling that the echo:
'Occupy' and 'space' here are not meant in their most general sense, but are mechanical terms usually reserved for creatures. So whilst the echo is not a creature, it inherits the capacity of creatures to occupy space, and the restrictions of space-occupation which come with them. Whilst these rules refer explicitly to creatures being unable to occupy one another's space, they are extended to the echo because the echo is a space-occupier even though it's not a creature. This is a difficult edge case brought about by poorly written rules.
What does this mean in practise?
Implications of this constraint
The echo cannot move through the space of a hostile creature (unless it's much bigger or smaller)
Creatures, which occupy space,
Since the echo occupies space, it follows that this restriction is applied to the echo's movement
The echo can move through a non-hostile creature's space (or the space of a much bigger or smaller hostile creature)
This is where it starts to get a little messy. The rules for players state that
Difficult terrain means double movement cost. The echo does not have a speed and does not use speed to move, so 'difficult terrain' is meaningless. Because:
The echo simply moves through the spaces of passable creatures without penalty
The echo cannot end its movement in another creature's space
This is where it really starts to break down. The echo does not expend movement, and so cannot 'end its movement' anywhere. I would argue here for what I think is the most intuitive ruling: The Echo Knight cannot stop moving their echo in the space of another creature, and is only permitted to move the echo through another creature's space.