I would guess that High Street in the (mostly British?) commercial shopping area sense would have evolved from “highway”, meaning a principal or main road, which in turn evolved from “high way”, being those roads constructed above grade, so that water would drain off the road into the adjacent ditches. The Romans [citation needed] tended to build all-wearher roads like this.

In American English, “highway” would be an odd term to apply to a shopping district – usually referring to a higher-speed road – but in some contexts, highway is understood to be any improved road. The California Vehicle Code uses this definition, so that “highway” basically means any public road.

At least in California, roads named High Street do exist, but don’t necessarily corespondent to being physically tall over its surroundings or other steets. If anything, a typical High Street is often the same in character as another town’s Main Street, which sort-of returns to the British meaning of shopping area again, at least in small towns.