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. In fact, 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 found this survey of semi-dualizing modules by Sather-Wagstaff: Semidualizing modules.

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.