Personally, I can't recognize specific fonts in the wild (especially Helvetica and its infinite lookalikes), but can certainly feel when a site uses an inappropriate font (usually too thin, but also things like using a UI font for longer-form text), to the extent I have a few userscripts that change some popular site's fonts to something more appropriate in my optinion.
That said, I once spent a couple weeks trying out various programming fonts/sizes, so maybe I'm an outlier. (I'm on Mac now, and the system Monaco font is fine. I don't remember what I had settled on when I was using Linux until 6 years ago)
I have to add that I recently upgraded from a 2k 27" monitor to a 4k one, and the first thing I noticed is how much nicer all the text is (again, macOS)
Other considerations are how does the font render g (is there a loop at the bottom, or just a tail) or a (does it have a "tail" at the top or is more like an o with an extra leg?).
People can get really worked up about these details, just like a coding style guide: if it differs from what you're expecting, it's distracting, but you get used to it after a few days/weeks.
Since last I went on this journey, I see that this wonderful website for comparing programming fonts popped up:
But more than noticing the differences between them, what's important is noticing when there's something wrong in a composition because of the font.
For example, if you show a UI that uses a serif font to someone, they'll notice there's something wrong with it, but most people probably won't tell you exactly what it is.