Not under most circumstances
The rules make no explict statements about what souls and spirits are, but from interactions with other game elements we can conclude that in their normal state, souls or spirits are not creatures. They can take the form of a creature under special circumstances.
First, what is a creature? The term is not explictily defined, but from the rules it is clear that it is anything with with game statistics (such as Intelligence, hit points, and AC), which includes player characters and monsters, NPCs being a subtype of monster.
Second, are soul and spirit the same thing? For the most part, the terms soul and spirit seem to be used interchangeably, soul is used more in the context of player characters and raising the dead, while spirit is used more commonly for nature and summoning. The speak with dead spell says
This spell doesn't return the creature's soul to its body, only its animating spirit.
so in this instance there is a difference between soul and spirit, but it seems that the term "animating spirit" means something different than spirit in general and is used more to convey limitations of the spell. So for this answer, we will consider soul and spirit as two different words for the same thing.
Souls or spirits certainly can be creatures. The Ghost, a medium undead creature has this lore:
A ghost is the soul of a once-living creature, bound to haunt a specific location, creature, or object
So she clearly is a soul and a creature at the same time. The Specter, another medium undead creature, has this lore:
A specter is the angry, unfettered spirit of a humanoid that has been prevented from passing to the afterlife.
Again, the specter is a spirit and also a creature. Likewise, multiple spells can summon spirits that take the form of creatures, for example Conjure Animals, Conjure Fey, Find Familiar, Find Steed. These spells typically have language similar to this from Conjure Animals
You summon fey spirits that take the form of beasts
Beasts, fey, etc. are all creatures, so all of these spells summon spirits that also are creatures.
You might call these creatures "manifested" souls or spirits, because they have taken form as a creature. In all of these cases, there is a special cause that made the soul or spirit take the form of a creature. What about the nature of the spirits before they took this form?
Souls like creaturs do have a defined location, typically because they are part of a creature, or entrapped in an object, or upon death (DMG p. 24):
When a creature dies, its soul departs its body, leaves the Material Plane, travels through the Astral Plane, and goes to abide on the plane where the creature's deity resides.
What about the other attributes required for creatures, like hp, and AC?
Spirit Guardians summons spirits, exhibiting angelic, fey or fiendish forms.
You call forth spirits to protect you. They flit around you
These spirits are clearly no creatures, having no AC, hp or way to even be attacked. This proves that generally spirits are not creatures, even when they do take visible form.
All instances of spells that talk about transferring souls (e.g. Magic Jar, Raise Dead, Reincarnation, Ressurection etc.), provide no game statistics for the soul, as would needed if the soul were a creature. Most of these spells also would not make sense, as a creature cannot "have" another creature.
From all of this evidence, we can conclude that in their default state, souls or spirits are not creatures under the game rules.
P.S. A special case is when a ghost takes Possession of another creature, the ghost then disappears but it retains its alignment, Intelligence, Wisdom, Charisma, and immunity to being charmed and frightened. It otherwise uses the possessed target's statistics. The two creatures (ghost and target) are somehow merged into one here, but in this case the manifested soul that is the ghost was a creature already to begin with.