Creatures have speeds, fly and otherwise. The Manifest Mind is not a creature; it's specifically an object that does some specific things. One of those things is move up to 30 feet at your command and hover in that space.
Nothing in the rule text says the space you move the Mind to has to be at ground level or a specific distance above the ground, and since it isn't using a movement speed, difficult terrain and similar effects relating to movement speeds don't matter. Moving the Mind around is similar to using the telekinesis spell to move a creature, or to blasting somebody through the air with thunderwave -- it's not flying and has nothing to do with the rules for flying, or even movement in general. It's just being moved around by an effect, in which case it does only what the effect says. Since the text doesn't specify "on the ground" or any such thing, it is not so limited.
Hover is not a game term, so we default to the standard english usage of the word. A hovering object doesn't fall under gravity. Put it wherever you want within 30 feet of where it was last time, even if that's straight up.
For comparison purposes, the Manifest Mind is very like a mage hand:
A spectral, floating hand appears at a point you choose [...] You can move the hand up to 30 feet each time you use it.
As with the Mind, there is no limitation on direction, no specification that it's flying, climbing, or walking. It says it's "floating" and "can move" but nothing more concrete than that. I suppose somebody could try to claim mage hand can only operate horizontally along the contour of the ground, but that seems fairly absurd.
For the contrary case, see the Artificer Artillerist's cannons:
...you can take an action to magically create a Small or Tiny eldritch cannon in an unoccupied space on a horizontal surface within 5 feet of you.
The cannon is a magical object.
...you can direct the cannon to walk or climb up to 15 feet to an unoccupied space, provided it has legs.
This is very specific about what the cannon can do: You don't "move" it, you
"direct" it to walk or climb, so it's limited by the rules of walking and climbing, even though it isn't a creature.