Let's analyze each statement.
Statement I: The test for the nickel(II) ion, Ni$^{2+}$, is a classic qualitative analysis reaction using an alcoholic solution of dimethylglyoxime (dmgH$_2$).
The reaction is carried out in a basic medium, which is typically achieved by adding ammonium hydroxide (NH$_4$OH). This results in the formation of a characteristic bright cherry-red precipitate of bis(dimethylglyoximato)nickel(II), [Ni(dmgH)$_2$].
Thus, Statement I is true.
Statement II: Dimethylglyoxime (dmgH$_2$) itself is a neutral molecule. However, when it coordinates with the Ni$^{2+}$ ion, each ligand molecule loses one proton to form the monoanionic ligand, dimethylglyoximate (dmgH$^-$).
It is this anionic form that coordinates to the metal. Each dmgH$^-$ ion acts as a bidentate ligand, coordinating through its two nitrogen atoms. Therefore, the ligand in the complex is bidentate but it is anionic, not neutral.
Statement II is false.
Since Statement I is true and Statement II is false, the correct option is (C).