Simply evaluating Element[\[Rho], Reals] does not make it into an assumption for Simplify. You can use $Assumptions = {\[Rho] > 0 && -Pi < \[Phi] < Pi} to make global assumptions.
Also, the assumption that rho is real is not enough to simplify the Abs to rho. You need to assume that rho is >0. Also, the Arg simplifies if you assume that phi is between -Pi and Pi:
FullSimplify[
Arg[\[Rho] Exp[I*\[Phi]]], \[Rho] > 0 && -Pi < \[Phi] < Pi]
