Yes, it applies to all movements
Unarmored Movement (PHB, 78) States:
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield.
It does NOT qualify as your Walking Speed (or Swimming Speed, or Flying Speed)...just Speed. This applies to all types of movements and would add additional speed to the Fly spell as well.
This is supported by Jeremy Crawford as well.
Unarmored Movement is intended to increase a monk's innate speed, including an aarakocra monk's flying speed.
He further clarifies Bonuses and Penalties that apply to Speed in general:
Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc
As the Unarmored Movement does not specify a specific type of speed, it applies to all.
Fly Spell specifically
Fly states (PHB, 243)
The target gains a flying speed of 60 feet for the duration.
The implication here is that you (the target) had no Fly Speed (or a lower Fly Speed) before casting and it is now a speed that you, the target have. Once it becomes your speed, it becomes subject to Unarmored Movement.
Similar to Special Types of Movement in the PHB, page 182)
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.
Here, Swimming has a penalty, unless they have a Swimming Speed. Once they have that Speed, from whatever source, then Unarmored Movement should apply.
Rule of Cool
And it's fun! Monks are supposed to be super quick. Let them be super quick in all their speed!
RAW is unclear, but likely not
Neither square or hover are clearly defined terms but from their context we can draw decent conclusions.
A hovering creature is still flying and would not be within the 2-D area of effect for entangle.
Entangle's Area of Effect
The area of effect for entangle is a 20-foot square. Square is not one of the shapes listed in the PHB section on areas of effect (PHB, p. 204). The DMG only mentions it as "Cube or Square" when determining the number of creatures within an area for gridless play (DMG, p. 249). Additionally, both the PHB and the DMG are fairly light on 3-dimensional rules like height for things other than cubes, spheres and cylinders.
In contrast the spell web has an area of effect of a 20-foot cube when anchored. Without anchors its area is describe as:
Webs layered over a flat surface have a depth of 5 feet.
We don't have a rule for how high the area of effect for entangle reaches, and therefore how high the demilich would need to be to avoid it.
In my interpretation, square is a 2-dimensional version of cube, therefore it has no height and a creature must be on the ground in the area to be effected. Therefore unless the demilich is touching the ground (which it doesn't need to) it should not be affected.
Hover ability
From the Flying Movement section under Movement and Position (PHB, p. 191) we have the meaning of "hover" (emphasis mine):
If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover [...]
The demilich's hover ability prevents it from falling when knocked prone, but otherwise has no effect on area of effect for spells like entangle. It does mean that the demilich can choose to be at any height without fear of touching the ground; however, other than that, it obeys the regular rules for flight.
To clear up confusion, hover is a more powerful version of flight, not less. Would you rule that a creature flying 5ft above the ground is effected by entangle? I would rule that it is not. So a hovering creature would not be effected either.
Best Answer
Speeds don’t determine this
That something has only a fly-hover speed and zero ground speed just means that it can’t move on the ground. It doesn’t say anything about whether it can or can’t stand still on the ground.
Speeds just aren’t there to give information on what media the creature is allowed to touch. If something is incapable of landing and never rests on the ground, that’s more the sort of detail that would appear in a creature’s detailed description of its behaviour and habits.
The demilich in particular lacks this kind of detail saying it only flies. Its description instead implies (but only implies) that it sometimes rests on the ground, when it says
Its history in D&D more explicitly includes it laying on the ground, including in one of its earliest illustrations in the AD&D Monster Manual II (and its accompanying description):
A classic demilich setup (as in the adventure
) is to have it laying on the ground until disturbed. To naive adventurers, it looks like a jeweled skull: treasure! Then it “wakes up” (it doesn’t actually sleep, though) and attacks, taking the poor adventurers off guard.