The term exists indeed, but I never heard it/used it outside continuum espace temps (and now I’m “back to the future” :p).

I would would go with your colleagues suggestion:

Mécanique des milieux continus

(I suppose there are several such domains, and not just one).

Here is the French wikipedia lemma for "continuum"

https://fr.wikipedia.org/wiki/Continuum

giving as an alternative

Le continuum mécanique

Sciences et technologies – Physique

Le continuum mécanique, une branche
de la physique

I don’t see anything faulty with continuum mécanique (grammatically speaking) other than its unknown usage. Of course it may sound awkward to French natives.
Obviously, Wikipedia is not panacea.

The term “continuum” is perfectly valid. Aside from the “continuum espace-temps”, which designates the four-dimensional space in which physics are set, there is the much less scientific expression “sur un continuum allant de … à …“, which in English would translate as “on a spectrum from … to …“.

I have never heard the plural form continua used.

As for the sentence that you were trying to translate originally, the correct translation is indeed mécanique des milieux continus. That’s an expression that maps directly to the concept of continuum mechanics in English and is used e.g. to describe a course in a curriculum.

Similarly Special relativity does not translate to *”relativité spéciale” but to relativité restreinte (“restricted“).