$*$ is certainly commutative. To check for an identity, we would need that there exists $a b$ such that
$$
a+b+2 a b=a
$$
for any $a \in \mathbb{Q}$. The choice $b=0$ works - it solves $b(1+2 a)=0$ for all $a \in \mathbb{Q}$. Finally, to look for an inverse, we would need a solution for $a * b=0$.
$$
a+b+2 a b=0 \Longrightarrow b(1+2 a)=-a \Longrightarrow b=\frac{-a}{1+2 a} .
$$
However, this only works if $a \neq-1 / 2$. If $a=-1 / 2$, then we would need to solve $-1 / 2+b-b= 0,$ which is impossible. Hence not every element has an inverse.
