I do not believe it is part of any existing lore.
It is hard to find evidence for the total absence of something, however I have never come across the term, and searching the PDFs I have accumulated from DMSGuild does not turn up any results. There is no mention of it on Forgotten Realms Wiki (one, two). I have not come across it while reading many novels set in the Forgotten Realms.
This verbiage appears to be new to 5e. Let's compare Speak With Dead across editions:
... This spell doesn't return the creature's soul to its body, only its animating spirit. Thus, the corpse can't learn new information, doesn't comprehend anything that has happened since it died, and can't speculate about future events.
You ask the corpse of an intelligent creature questions and receive answers. The corpse knows what the creature knew in life, what has occurred near the corpse, and no more; the spirit has (usually) moved on to another plane and is not present in the body.
You grant the semblance of life and intellect to a corpse .... The corpse's knowledge is limited to what the creature knew during life .... Answers are usually brief, cryptic, or repetitive .... This spell does not let you actually speak to the person (whose soul has departed). It instead draws on the imprinted knowledge stored in the corpse. The partially animated body retains the imprint of the soul that once inhabited it, and thus it can speak with all the knowledge that the creator had while alive. ...
... the priest is able to ask several questions .... Even if the casting is successful, such creatures are as evasive as possible when questioned. The dead tend to give extremely brief and limited answers, often cryptic, .... A dead creature that successfully saves can refuse to answer questions ...
... the cleric is able to ask several questions of a dead creature .... The length of time the creature has been dead is a factor, since only higher level clerics can converse with the long-dead ...
Basic/Expert - not present
A spell which allows the user to converse with a dead body, the Cleric being able to ask three questions. Note that the length of time the creature has been dead will be a consideration. ... riddles [are] recommended! ...
Looking back through these incarnations, it looks as if this is just verbiage to illustrate the unhelpfulness of your conversational participant cough
There is no official D&D cosmology, and if you try to construct one, you will find books published in the same year directly contradicting each other. Cosmology is ultimately up to you.
On page 12 of the DMG, during a discussion of Animism, it is stated,
Animism is the belief that spirits inhabit every part of the natural world. In an animistic worldview, everything has a spirit, from the grandest mountain to the lowliest rock ...
So, sorry, but there do not appear to be any other references to "animating spirit", anywhere.