Actually this is the same meaning as in the question you referenced. More precisely it is the same as the third sentence:

Ces gens vendraient leur âme pour avoir ne serait-ce qu’une bribe d’information sur l’incident.

Your sentence is slightly more complicated because it is already in the negative form, but it still works the same.

Your question is based on the incorrect assumption that this French sentence contains a double negative while it doesn’t.

There is only a single negation “aucun xxx ne xxx” and no negation but a restriction in “ne xxx que“. A restriction is limiting a magnitude, it doesn’t reverse anything.

A real double negative might have been:

Aucun ne parlait même pas un simple mot de mandarin.

but this sentence is confusing and incorrect in French too.

You also dropped several ellipses present in the original sentence which reads:

Ludovic : Ah oui, je me souviens… je me souviens aussi des cours de chinois, j’ai un petit… j’ai notamment un souvenir… donc pareil, on avait un… un journal de bord. Ou on était peut-être plus… enfin dans mes souvenirs, on était plus focalisés par exemple sur ce que nous on ressentait mais après… enfin, on regardait aussi un petit peu la technique de… de l’enseignante puisque donc tout… le cours en entier se faisait en chinois, enfin en mandarin.

Jessica : Ouais.

Ludovic : Et le… le problème c’est que… bah forcément aucun de nous ne parlait un… ne serait-ce qu’un mot de… de mandarin.

Jessica : Hm hm.

So the originally intended sentence was:

aucun de nous ne parlait un mot de mandarin

i.e. none of us spoke one word of Mandarin.

but in the middle of the sentence, the stuttering speaker replaced it by:

aucun de nous ne parlait ne serait-ce qu‘un mot de mandarin

Here, an intensifier idiom has been inserted into the sentence (it is an incise). It says ne serait-ce qu’un mot, i.e. should it be just one word / not even a single word so the full sentence might be translated to:

None of us spoke one… even a single word of… of Mandarin.

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Another fact that might be confusing is that the first ne is not per se a negative mark but part of the split negative aucun … ne. There is no extra negative here so

Aucun de nous ne parlait…

really means, as already stated:

None of us spoke…

and not

None of us didn’t spoke…

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Moreover, the second ne is part of fixed French formal adverbial phrase ne serais-ce que which technically just means even¹ so as already stated, there is no double negative either (i.e. a combination of negations that lead to a logically positive sentence, like I don’t know nothing about grammar), and even less a triple negative.

The sentence can then simply be rephrased as:

None of us spoke Mandarin, even a single word of it.

¹_{gabrielwiler.com states: On pourrait aussi dire que « neseraiske » est un adverbe de perspective proche de même.}

‘Not even’ is the closest translation in this context.

I’ll try my go at an answer, mostly to try and distribute the comments a little and have more space to clarify my point.

First thing, as Jlliagre said, your sentence is not correct. Maybe it was said orally with exactly these words, but you can’t write it that way with no punctuation. At least maybe you should mention somewhere that it’s from an oral conversation?

What do you mean by No, that is the correct amount of 'un's? The first one should just not be there.

From now on I’ll assume the sentence is as follow:

Aucun de nous ne parlait ne serait-ce qu’un mot de la langue.

Obviously your question is not about the meaning (you guessed it right and you said that it’s clear what it means from context), but about why it means what it means.

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The answer to the linked question said “in a negative context, meaning differs slightly. It is synonym of même pas (“not even”)“, and I don’t entirely agree with this. I think it’s misleading.

To me the meaning is more or less the same, it’s just that is translate differently. You can replace it with “juste” in practically every sentence and still convey the same meaning (slightly less extreme though):

(It sounds a little awkward but the meaning is there so it’s not that different from a context to another.)

J’ai du mal à faire confiance aux étrangers, juste à cause de ces histoires d’intrus.

= This “intruders thing” is enough to make it hard for me to trust strangers

Vous allez accepter, de toute façon, juste pour l’or que vous rapportera cette mission.

= Just for the gold, you’re going to accept this mission anyway.

Ces gens vendraient leur âme pour avoir juste une bribe d’information sur l’incident.

= These people would sell their soul to have even one tiny piece of information about the incident

Aucun de nous ne parlait juste un mot de la langue

= None of us spoke a single word of the language

If you’re worried about the double negative, the meaning is exactly the same with full positive:

J’aimerais parler juste un mot de la langue

= I wish I spoke just a single word of the language

If it’s still not clear feel free to ask any question you have, at least I can edit the answer.