Can a physical quantity have units but still be dimensionless?

yes

Explanation:

there are other quantities that are dimensionless, but have a unit.

For example, radian is the unit for an angle. But it is dimensionless because it is defined to be the ratio of two lengths.,

a dimensionless quantity will always be independent of the base units — meter, second, kilograms, kelvin, candela, moles, ampere.

Other examples:

1. Mass fraction

2.Mole fraction

both have unit (kg/kg)

elctricity and magnet is conversion to it