New magic theory based on Idia mentioning the memory wipe and alternate river styx. Skip to the end for my answer.
I thought they were going do the thing in Men in Black by using something similar to the neutralizer to erase everyone's memories (y'know like the stature of liberty) but perhaps they memory wipe using something like Idia's powers. Using Ortho as our example, let's take a look.
We still aren't given a direct answer to what Ortho is (as far as I know, correct me if i'm wrong), we just know that he is Idia's brother who might be a cyborg or a straight up android with no human components.
However if he is actually a cyborg then it could possibly mean that with the help of Idia's magic he could become the Ortho we know, a machine with Ortho's memories.
Now back to how this relates to the river and memory wiping.
Idia mentions that their memories will be 'floating down the Lythe river' and with the concept that people are what they make of themself, their memories make up who they are as a person as they themself are influenced by the world around them. Y'know, the whole reason all our boys got overblotted in the first place because of negative memories/influences. Also adding that the human body also runs on memories i.e muscle memory, involuntary movement run by the brain's memory of what to do, stuff like that.
So, taking their memories is basically the same as 'taking their souls' except in this context, they are only taking a bit of what they remember at STYX and 'sending it down the Lythe river' to be forgotten (like many eventually in the time of death which is why humans make memorandums of people) which could be a database in the context of technology (looking at the binary code of people in Idia's and Ortho's dorm art)
By jumping the gun here, it's possible, with another reference to the Ignihyde anime like intro, showing Idia placing the likely 'memories' into Ortho-
-or in this case binary code now, that those are the 'souls/memories'. Which in a way also means they that they either had a 'backup file' for him OR Idia is using his memories of his brother to bring him back. This could explain the 'two Hades' thing as Ortho is actually a piece of Idia's soul/memories rather than being his actual brother (like the Leech twins).
As Vil mentions, even if it was more of a line of humour, that Ortho is just like his brother, when he mentions that them being dragged here all of a sudden will affect the results of the test rather than being in actual consideration of the morality of it.
And with Idia being haunted by souls, also in the intro...
It's possible that it really is his unique magic as he gets pulled/weighed down by the 'souls' he had taken (or the boy is lowkey traumatized/haunted by the concept of people's souls living forever in a computer).
The ability to remove or add souls/memories. It could make you forget how to fight, your passions, your training, who you are as a person or even how to breathe. It may also be the reason for the line 'those who go to STYX never return'. They don't return as the same person they were when they entered.
A scary magic if you think about it.
It seems in line with Hades being the God that manages the wayward souls that end up in the river Styx. Like the tale of Orpheus and Eurydice, or more Disney context, Meg and Hercules, in order to return the soul to the living it comes with a small trick.
Don't look back and have trust that she is there or she goes back and Dive into death's realm to get her back but you could die, he forgot to mention that.
For Idia it was, get your brother back but he isn't.
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