Question is simple. If an Echo Knight/Barbarian uses Reckless Attack on a creature and hits, will attacks made against their echo also be made with advantage?
The "Reckless Attack" barbarian class feature says:
Starting at 2nd level, you can throw aside all concern for defense to attack with fierce desperation. When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn, but attack rolls against you have advantage until your next turn.
Now lets read about echo knight.
Your echo has AC 14 + your proficiency bonus, 1 hit point, and immunity to all conditions. If it has to make a saving throw, it uses your saving throw bonus for the roll. It is the same size as you, and it occupies its space.
When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo’s space. You make this choice for each attack.
So, in this situation, there are a couple of key points of information. It is its own entity when it comes to being attacked. It has its own AC and HP. However, it is incapable of making its own attacks. Instead YOU make attacks originating from it's space.
This means in the case of reckless attack, it grants advantage to attacks made from the echo's location (because you are making them, just originating from another location), however it does NOT grant advantage on attacks against the Echo, because it is its own entity.
One additional note: I used the word entity here, because the ability never specifies that the echo is an object or creature. Compared to say Artificer subclass Artillerist which clearly identifies your cannon as a magical object.
The cannon is a magical object.
This means that very strictly RAW, the echo cannot be targeted by many spells, which require a valid target (typically a creature or object), where weapon attacks can target a location. It also would make it invalid target for Opportunity attacks which require:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
I would suggest DMs treat this as an oversight and allow Echos to be treated as creatures, objects, or both.
This echo is a magical, translucent, gray image of you.
It is only an image of you.
The Reckless Attack feature includes the following line at the very end of its description (emphasis mine):
attack rolls against you have advantage until your next turn.
Since your echo is decisively not you, Reckless Attack does not grant enemies advantage when attacking your echo, only against you.