With the current configuration Prettier seems to be installed on every run - which can been seen from the output:

npx: installed 1 in 5.166s

Prettier isn't explicitly being installed (it's surprising that actually installing the dependencies isn't included in the actions/cache docs) but it turns out that npx will automatically install the package for the specified command (it actually guesses the package name from the name of the command). I'm not sure where Prettier ends up being installed but it doesn't appear to be in ~/.npm according to the post-cache output (or ./node_modules when I tested locally):

Cache hit occurred on the primary key Linux-npm-565329898f77080e58b14d45cf816ab94877e6f2ece9d395c369c533548a7ee7, not saving cache.

I think there are a couple of approaches to tackling this, you could manually install/cache Prettier within the action, or add a package.json with Prettier. I would go with the latter because it's a more standard and maintainable approach and it will also ensure that, along with CI, anyone working on the project will run the same version of Prettier (you'll also get Dependabot JavaScript updates).

I've tested the package.json approach on a branch and am happy to turn it into a pull request if you fancy.