Group: For any algebraic structure to be a group, that has to satisfy the Closure, Associatively, Identity and Inverse properties.
Closure
For all $a, b \in G$, the result of the operation, $a * b$, is also in $G$.
In above example, Since there is one element, hence $ a = b = 0 $, and $ a * b = 0 * 0 = 0 \in G $
Hence Closure satisfy.
Associative
For all $a, b, c \in G$, $ (a * b) * c = a * (b * c) $.
For above example, $ a = b = c = 0 $
Hence $ (a * b) * c = a * (b * c) $
$ \implies (0 * 0) * 0 = 0 * (0 * 0) \implies 0 = 0 $
Hence Associatively satisfied.
Identity element
There exists an element $e \in G$ such that, for every element $a \in G$, the equation $ e * a = a * e = a $ holds. Such an element is unique, and thus one speaks of the identity element.
For above example $ a = e = 0 $
Hence $ e * a = a * e \implies 0 * 0 = 0 * 0 \implies 0 = a $
Hence $ e = 0 $ is the identity element.
Inverse element
For each $a \in G$, there exists an element $b \in G$, commonly denoted $a^{−1}$, such that $a * b = b * a = e $, where $e$ is the identity element.
For your example, The inverse element is $0$. Because when you multiply $0$ with $0$ then you will get $0$, which is also an identity element of the structure.