Independent Axes

A thrown ball is two motions at once: sideways along x and up and down along y. Nothing pushes sideways, so the horizontal velocity never changes. The vertical motion is free fall:

$$x = v_x t, \quad v_y = v_{0y} - g t$$

The two motions share only the time $t$: dropped and thrown sideways from the same height, two balls land together.

Same height at the same instant: the sideways motion does not change the fall.