An animating spirit is likely the intangible foundation of a creature (other than certain constructs that rely on animating magic). Accordingly, an animating spirit is in effect a creature, likely has zero hit points, and is likely not a monster.
As an aside, it appears likely that every soul has an intangible spirit. Refer to the description of Revenant in MM pg 259 which indicates:
A revenant forms from the soul of a mortal
and later in the description:
If the revenant's original body was destroyed or is otherwise
unavailable, the spirit of the revenant enters another humanoid
Accordingly, a Revenant (like a Wraith) has an animating spirit, however it likely does not have a soul (a Wraith definitely does not have a soul). This is different from a Specter (a spirit that does have a soul) and a Ghost (a soul, presumably with an animating spirit).
There appear to be two types of spirits:
intangible spirits and incorporeal spirits.
An intangible spirit does not create difficult terrain for other creatures moving into or through it's space. Other creatures can move into it's space without it being slowed, and can end their turn in the intangible spirit's space. Refer to the Healing Spirit spell in XGtE pg157 which refers to the spirit as "intangible".
An incorporeal spirit has a Fly (Hover) speed and it can move through other creatures and objects including hostile creatures as if they were difficult terrain, and can end its turn in another creature's or object's space. However, it will take force damage if ends its turn in another creature's or object's space. Refer to the descriptions of various undead in MM including Ghost, Specter, and Wraith.
Spirit Guardians from the spell of the same name in PHB pg278 appear to also be incorporeal spirits in a reverse fashion, as hostile creatures appear to be able to move through their space as though it were difficult terrain, and it it the hostile creature that takes damage (Radiant, rather than force, in this case) when, among other things, finishing it's turn in that space.
An animating spirit, while being a creature, is likely not a monster. The definition of monster in MM pg 4 indicates the following (bold emphasis mine):
A monster is defined as any creature that can be interacted with
and potentially fought and killed.
From this, it follows that an incorporeal spirit such as a Specter is clearly a monster.
However, it is does not appear that an intangible spirit like a Healing Spirit is a monster, as it does not appear that it can be fought or killed. It appears likely that an animating spirit or the spirit from a Find Familiar spell would be similar.
Finally, I would argue that it is likely that intangible spirits have 0 HP. From the descriptions of Polymorph and True Polymorph spells in PHB, it is clear that incorporeal creatures can not be polymorphed, and it seems nonsensical that intangible creatures could be polymorphed. Instead, those spell make clear that creatures with 0 HP cannot be polymorphed, which to me strongly indicates that - along with the limited interactions ascribed to intangible spirits - intangible spirits have 0 HP.