If the mist settles over Lake Ashi at Hakone, does the reflection of Mount Fuji in the water happen at the same depth as the mountain’s actual height, or does the water level dictate the scale?

Reflections are light bouncing off the surface of the water, so it’s more to do with angle (which is influenced by distance)

https://ranchocordovaarts.org/art-tips/water-reflections

The object and its reflection will have an identical height only if your direction of view is nearly parallel to the water surface.

In your example, the photographer is in a position where the two are equal. The characteristics of the water are not relevant.

The reflection happens at surface level and it doesn’t have “depth”, so the water level dictates that. The actual light dynamics are beyond my knowledge