= Solution
Yes, but there is only one nontrivial isomorphism type that works. If $G=C_2$, then
$$
G*G=C_2*C_2\cong D_\infty.
$$
The <infinite dihedral group> contains its infinite cyclic rotation subgroup with index two, so it is amenable by <virtually abelian groups are amenable>.
Suppose now that $G\not\cong C_2$. The action of $G*G$ on its <Bass-Serre tree> is non-elementary: one vertex degree is greater than two, so there are hyperbolic elements with disjoint pairs of endpoints. Suitable powers satisfy the <ping-pong lemma> on the boundary and generate a copy of $F_2$. Since <a subgroup of an amenable group is amenable>, an amenable group cannot contain this nonamenable subgroup. The <amenability of a free product> therefore yields
$$
\boxed{G*G\text{ is amenable}\iff G\cong C_2}
$$
for nontrivial finitely generated $G$.
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