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I’m not sufficiently familiar with French culture or maths culture (and barely at all with their intersection!) to tell whether this is satire/parody, or just bonkers in general. Which is it?

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It's a satire of a typical kind of paper from logic, in particular modal logic. Jean-Yves Girard has been very vocal against these academic papermills where the authors consider ad-hoc meaningless logical systems. For a more in-depth critique of semantics, you can also read the broccoli logic paper: https://girard.perso.math.cnrs.fr/meaning1.pdf

As I understand it (also neither a mathematician nor French), it's a parody on a certain type of math paper, I'm guessing in abstract algebra, where the author does nothing but present a theory that is a "generalization" of some other theories, with no concern whether this generalization is helpful in any context.

So you get a "mustard watch", a thing which behaves simultaneously as a watch and as a mustard container, and finding a situation where that would be useful is an exercise for the reader.

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