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!