Does circulating charge require both angular momentum and magnetic? Consider positive and negative charges simultaneously circulating and counter circulating.

Electrons flow from the negative pole to the positive pole. Conventional current or simply current behaves as if positive charge carriers cause current to flow. Conventional current flows from the positive terminal to the negative terminal.

Positive and negative charges are simultaneously circulating and counter-circulating.

If positive and negative charges move in circles in the same xy plane, the circulation of the positive and negative charges is opposite. Suppose both positive and negative charges have the same mass. In that case, their total angular momentum could be zero because the angular momentum of each would be in the opposite direction along the z -axis.

No, the circulating charge does not require both angular momentum and magnetic moment.