RPGMemes

D&D's targeting rules are quite strange, but yes, it's very explicit that Chebyshev is used in 5e by default, if playing on a grid. On page 192 of the 5.0e PHB:

To enter a square, you must have at least 1 square of movement left, even if the square is diagonally adjacent to the square you're in.

The DMG presents, on page 252, an optional variant of the optional grid rules, which is to treat it the same as Pathfinder 2e does (alternating 5 ft and 10 ft):

The Player's Handbook presents a simple method for counting movement and measuring range on a grid: count every square as 5 feet, even if you're moving diagonally. ... This optional rule provides more realism.

When measuring range or moving diagonally on a grid, the first diagonal square counts as 5 feet, but the second one counts as 10 feet. This pattern...continues when you're counting diagonally even if you move horizontally or vertically between different bits of diagonal movement.

As for the value of cube vs sphere in the context of Chebyshev ranges, there are two key differences.

First, cubes measure side length, spheres measure radius. A 10 ft cube covers 4 squares. A 10 ft sphere covers 16.

Second, and more importantly (since the above could easily be translated by using only cubes or only spheres throughout the system, with either half or double the numbers), cubes are cast from one side, whereas spheres are cast from the centre. If you're standing in the front line with enemies in front of you and allies behind, a cube cast with you as its origin point will hit either allies only or enemies only, but not both. A sphere cast with you at its origin point will affect both allies and enemies. Note that the rules for cube, on page 204 of the 5.0 PHB say "A cube's point of origin is not included in the cube's area of effect, unless you decide otherwise." So you could include yourself and your allies, or you could include enemies but not yourself, if you so desired. Or, less likely, you could include allies but not yourself, or enemies and yourself.

From memory, cube spells are mostly cast from a range of "self", which is where this becomes an important distinction. If a spell has a range of X feet and cube, then the main difference is just that its area is smaller but its reach is longer than a sphere with the same numbers.