In fact, the FR team was specifically told that FR characters could not be any more powerful than a similarly equipped core D&D character. Yes, regionally-appropriate characters start with about 300 gp of gear more than core characters, but after a level or two that difference is negligible and all the extra gear means is the character is more likely to survive in that critical 1st-2nd level range.
So according to the rules, FR is not high magic. The themes of the setting usually rely on magic, but there's nothing hard-coded into the system to make you play it that way. It uses the core rules.