All classifiers in scikit-learn do multiclass classification out-of-the-box. You don’t need to use the [`sklearn.multiclass`](classes.html#module-sklearn.multiclass"sklearn.multiclass") module unless you want to experiment with different multiclass strategies.
At present, no metric in [`sklearn.metrics`](classes.html#module-sklearn.metrics"sklearn.metrics") supports the multioutput-multiclass classification task.