Perhaps I can say that finitely generated modules for which
$$R \to \mathrm{Hom}_R(M, M) \;\;\;\text{is an isomorphism, where $1 \mapsto id$}$$
and
$$\mathrm{Ext}^i_R(M,M) = 0 \;\;\;\text{ for $i > 0$}$$
are called semi-dualizing modules (if they are in $D^b_{coh}(R)$, they are called semi-dualizing complexes).
This can be more compactly written as
$$R \to {\bf R}\mathrm{Hom}_R(M, M)$$
is an isomorphism.
The point is that $M$ would be a dualizing/canonical module (respectively complex) if it had finite injective dimension.
There's actually a lot of work on identifying semi-dualizing modules/complexes, and they are relatively rare. For a Gorenstein local ring $R$, the only semi-dualizing module is $R$ itself up to isomorphism (in fact, the only semi-dualizing complex is $R$ up to shift). See Corollary 8.6 in Christensen, Semi-dualizing complexes and their Auslander categories. For a Gorenstein variety, it follows that the only semi-dualizing modules are line bundles.
If you weaken the condition of Gorenstein to Cohen-Macaulay, then there can be more. If I recall correctly, there are still only finitely many, and there's an even number (see Christensen-Wagstaff). In fact, unless things have changed in the past few years, no one knows an example where the number of semi-dualizing modules/complexes is not $2^n$ for some $n$.
I also found this somewhat older survey of semi-dualizing modules by Sather-Wagstaff: Semidualizing modules.