Fascinating. Maybe the next step will be to extract the tokenized interlingua language that's emerged in the neural network and map it to real words, and blam, we reinvent Esperanto!
In all seriousness, however, different languages often have radically different concepts encoded into them. For example, in English sentences you might be forced to explicitly give subjects to the actors in your sentences, assign genders to your pronouns, and choose whether actions take place in the present or future, whether they are ongoing in general or happening at this moment, etc. In Japanese you might be forced to choose how to express your relationship with the person you are speaking to, the person you are speaking about, and decide whether a causal relationship is "if and only if" or just plain "if then".
There's long been a hypothesis about whether there is some kind of "universal grammar" (Chomsky) but modern linguists do not really entertain that idea.
I know there are some possible counter-examples to universal grammar, like Dan Everett's work on Pirahã, a language which may lack recursion, and that there's more interest in usage-based theories recently. However, my impression is that most linguists are still pretty attached to a some sort UG-like theory of language and language acquisition. (Admittedly, this is partly because no one can agree on what does or does not constitute UG, but still...)
There have been a bunch of pop-sci articles about non-Chomskian points of view lately, but I'm not sure how well (e.g.) Tom Wolfe is plugged into theoretical linguistics gossip and a lot of the others seem to draw from the same pool of authors, like Paul Ibbotson, who had a piece in Scientific American and another in Salon.
I have not really been plugged into the pop science side of linguistics, so I'm not really familiar with any of the articles in Scientific American or Wolfe's recent work. However, I don't think that many people in linguistics have taken universal grammar very seriously since the 1970s or so, but that might just be my view of things.
Chomsky's ridiculous and untenable ideas about a "language organ" can't be taken seriously. Anyone with a defensible stance on UG therefore defines UG as something else, and whatever that "something else" is ends up not quite useful enough to actually hang on to. "UG-like language acquisition" seems like a bit of a contradiction to me, since the original concept of UG is of an innate grammar (and if you have to acquire it, it's not innate).
Perhaps these anti-Chomsky pieces get so much circulation because firstly, they're easy since Chomsky's old theories are practically indefensible and long out of fashion and secondly, Chomsky's a household name so you can move advertising dollars by picking on his old ideas. There will always be a market for articles that tell readers that some famous intellectual was wrong once.
One interesting thing that might still be left is criteria about what languages would be learnable or unlearnable. For example, perhaps there are reasons that it's impossible for human beings to become fluent in a conlang like Ithkuil (or Lojban), even if they were exposed as children to interactions with (hypothetical) fluent speakers. Yet this isn't true of Esperanto (or at least very slightly creolized versions of it), which does have native speakers for whom the language didn't seem to present any challenge. There could be criteria about what makes a language too weird or too hard to learn and speak with the facility with which all existing natural languages can be learned and spoken by almost all children; maybe some of those criteria are fuzzy but others are firm. I think this is related to UG, but clearly a weaker concept.
For instance, as you've said, Japanese encodes the relation to whom you're speaking. Even French does this (to a lesser extent), with "tu"/"vous" depending on how familiar the person you're speaking to is.
https://en.wikipedia.org/wiki/T%E2%80%93V_distinction