What is the unit of Z effective?

This is an example of terminology which should be taken with a grain of salt.

The term "effective nuclear charge" is often casually symbolized with $Z_mathrm{eff}.$ This is a universally accepted simplification, but you should keep in mind that the effective nuclear charge is, strictly speaking, $Z_mathrm{eff}e,$ where $e$ is the elementary charge. Similarly, the full nuclear charge is $Ze,$ where $Z$ is the atomic number.

So, it would be more appropriate to refer to $Z_mathrm{eff}$ as to effective atomic number since $Z_mathrm{eff}$ is defined as a function of $Z$ $(Z_mathrm{eff} = f(Z_i)).$ For example, the common form provided in every introductory physical chemistry textbook is

$$Z_mathrm{eff} = Z − σ,$$

where $σ$ is the shielding constant of the nucleus (empirical dimensionless parameter).

Since $Z$ is dimensionless, so is $Z_mathrm{eff}.$ However, effective nuclear charge $Z_mathrm{eff}e$ is not, and can, for instance, be expressed in coulombs (in the SI system $e = pu{1.602176634E−19 C}).$