Surely we have “car highway classification” because it exists in the real world outside OSM? Where I am in the UK we have public roads that are motorway, trunk, primary, secondary and unclassified; various oddities that tend to get mapped as track, and private or permissive service roads. Of the OSM “car roads”, only tertiary and residential are actual OSM inventions.
We don’t have, and I can’t think of anywhere that does have, a similar “pedestrian highway classification” (although some places do have something a bit like that for bicycles - at least in name).
Surely not - the router or map that you’re looking at does that for you.
So make your own map? An example schema (that I use for both raster and vector maps) is here. It separates paths into several categories. Ignoring the ones based on designation (likely not relevant to you if you’re outside England and Wales), you can see a split between pathwide and pathnarrow, and between good, not-good, “intermediate trail_visibility” and “bad trail_visibility” (actually a selection of tags is used to determine what bucket things go in). If you want to change the colour of “good narrow paths” from black to puce you just need to edit this line in the vector style to change it it vector maps. If the schema already has the data in it you don’t need to make new vector tiles, just edit one text file.
Outside of the UK, there are many endless discussions about “is this road a trunk or not” - often it’s OSM class comes out of its physical properties! There are also plenty of places (perhaps you remember this one) where the official classification is so far adrift of the road quality that the OSM class has been all over the place.
Is there an implied quality to any of those in your dialect of English? There mostly isn’t in mine; the exception maybe is “mutli-user path” which implies that it’s usable by at least some people on wheels (so likely a lower tracktype, a better smoothness and a better surface). A “rail trail” is likely fairly flat, but other than that you’d struggle to assume anything. The others just mean “some sort of path” in English English.