Souls and spirits per se are not creatures
The rules make no explict statement on this, so there the DM should have the final word, but from interactions with other game elements we can conclude that normally, souls or spirits1 are not creatures in the game's sense. They can take the form of creatures under special circumstances.
Un-manifested souls and spirits
Souls and spririts generally lack the statistics required for creatures if you want other rules to interact with them, including attributes, hp, or AC. There is no meaningful way to target something without AC with an attack, or something without attributes with a spell that has requires a save.
Spirit Guardians summons spirits, and 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. In the context of these spells souls as creatures also make no 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.
Manifested souls and spirits
So what about creatures like the Ghost or Specter, wich are described as souls? Both of these are monsters and creatures. The Ghost, a medium undead creature, has this lore:
A ghost is the soul of a once-living creature, (...)
The Specter, another medium undead creature, has this lore:
A specter is the angry, unfettered spirit of a humanoid (...)
Again, the specter is a spirit and also a creature. Lore text is not fluff, it is information for the DM in language that the characters in-universe can understand. So the ghost and the specter are souls and creatures at the same time.
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.
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.
1 As an aside, are soul and spirit the same thing? For the most part, the terms seem to be used interchangeably, soul more in the context of player characters and raising the dead, spirit more in the context of nature and summoning.
The speak with dead spell differntiates them when it 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.