In this lectures notes Geometric wave equation by Christian Bär at page 122 he has
Definition 3.4.1. Let $M$ be a timeoriented connected Lorentzian manifold. Let $P$ be a normally hyperbolic operator acting on sections in a vector bundle $E$ over $M$. A linear map $G_{+}$: $\mathcal{D}(M, E) \rightarrow C^{\infty}(M, E)$ satisfying (i) $P \circ G_{+}=\mathrm{id}_{\mathcal{D}(M, E)}$, ....
Here $C^{\infty}(M, E)$ is the space of infinty differentiable section and $\mathcal{D}(M, E)$ is the space of compact supported section for the vector bundle $E\rightarrow M$
What is the section $\mathrm{id}_{\mathcal{D}(M, E)}$?
My first answer was wrong, I was misled by the words in your final question, referring to $\mathrm{id}_{\mathcal{D}(M, E)}$ as a section.
Each individual element of $\mathcal{D}(M,E)$ is section of the bundle $E \mapsto M$, and $\mathcal{D}(M,E)$ is the space of all sections.
And as you noticed in the comments, $\mathrm{id}_{\mathcal{D}(M, E)}$ is a certain operator on $\mathcal{D}(M,E)$, i.e. it inputs a section and outputs a section.
Namely, $\mathrm{id}_{\mathcal{D}(M, E)}$ is the identity operator, whose output equals its input: $$\mathrm{id}_{\mathcal{D}(M, E)}(\varphi)=\varphi \quad\text{for all $\varphi \in \mathcal{D}(M,N)$} $$
So what that equation $P \circ G_{+}=\mathrm{id}_{\mathcal{D}(M, E)}$ says is that $G_+$ is a right inverse for the operator $P$ with respect to operator composition: When $P$ is composed in advance by $G_+$, the result is the identity operator.