The scale of a grid map is generally the distance between the centers of immediately adjacent cells, and in particular this is the case for the hex maps recommended by 5th Edition Dungeons and Dragons for overland travel. This is so that travel distances can be calculated by multiplying the number of cell-to-cell transitions by the map scale, or conversely that the number of cells to be traversed can be calculated by dividing travel speed by the map scale. This is not stated explicitly in the Dungeon Master’s Guide in the section on mapping, but can be inferred from these passages on page 14, “Mapping Your Campaign.” First, from the section “Kingdom Scale”:
On a kingdom-scale map, each hex represents 6 miles.
Then from the section “Continent Scale”:
For mapping a whole continent, use a scale where 1 hex represents 60 miles.
And finally from the section “Combining Scales”:
Two cities that are 3 hexes (180 miles) apart on your continent map would be 30 hexes apart on your kingdom map, and might define the opposite ends of the region you’re detailing.
From this we can infer that one can calculate the distance between hexes by multiplying the scale by the number of hexes traversed, which is only true if the scale measures the distance between centers of adjacent hexes.
For a square grid, this happens to also be the length of the edge of a grid cell. This is not the case for other regular divisions of the plane.