Never tried it, but I believe that you need to encode the language into the filename of the template, see this page for examples: Template lookup order | Hugo (The Language section near the top, and the example tables in the second part of the page, they have example template names for a French site)
(side note: I was gonna say I have already tried above solution and failed. BUT, once I delete public folder and run hugo server again, then it worked. I have been confused with other stuff as well but turned out hugo server did not update the site as I change stuff even when I terminate it and run hugo server again. Even with hugo serve -D --cleanDestinationDir --disableFastRender, it requires manually deleting public. Is this an expected behavior? I made sure it is not browser cache problem.)