In RAW, Mage Hand is "A spectral, floating hand appears at a point you choose within range".
But what is it "floating"? Can hand fly up to 10-20 feet over the floor? With payload (like small stone)? Can it fly up or down by command?
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To float generally means stay in place while not touching the ground, being supported, or suspended. As the spell describes the ability for the hand to move between two points and remain floating, that implies that the hand can also fly.
In past editions there was a distinction between float/hover and flying, as to hover while flying required a skill check, or there were abilities that let you float but not fly.