Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that $$ Mod(Th(A))=Var(A)? $$ Clearly finite algebras do not have this property. It seems that such an algebra should be relatively free.

This question is related to my previous two questions

1. Relatively free algebras in a variety generated by a single algebra

2. relatively free groups in $Var(S_3)$