This is a theorem of Rosenstein from the paper $\aleph_0$-categoricity of groups. Often the terminology "finite order" means "finite cardinality" in the setting of groups, so to avoid confusion I will borrow Rosenstein's terminology instead: A group $G$ has bounded order if there is some integer $n$ such that $g^n=1$ for all $g\in G$.
I also say that a group is $\aleph_0$-categorical if its complete theory in the language of groups is $\aleph_0$-categorical (i.e., has a unique countable model).
It is not hard to show that an $\aleph_0$-categorical group has bounded order. This is also Theorem 1 of Rosenstein's paper. Theorem 2 of his paper provides a converse for abelian groups. So we have:
Theorem. (Rosenstein 1971)
An infinite abelian group is $\aleph_0$-categorical if and only if it has bounded order.
As a side note, I had not heard the terminology "totient group" used this way before.