Given that $:a\square b=ab+(a+b)$
Identity element of this operation, defined as the number $x$ such that $a\square x = a$
$\implies ax+(a+x) = a$
$\implies ax+x=0$
$\implies x(a+1)=0$
$\implies x =0,a=-1\in \mathbb{R}$
So, the correct answer is $(A).$