Yes, it is disappointing. This gets rid of Arg in terms of ArcTan:

ComplexExpand[Arg[x + I  y] - 2  ArcTan[y/(x + Sqrt[x^2 + y^2])], 
 TargetFunctions -> {Re, Im}]
FullSimplify[%, x \[Element] Reals && y \[Element] Reals && y != 0]

How did you type the squares? The full form of your x^2 gives x\262. The two expressions are not totally equivalent on the negative real line:

{Arg[x + I  y], 2 ArcTan[y/(x + Sqrt[x^2 + y^2])]} /. {x -> -1, 
  y -> 0}

I suppose the two functions Arg and ArcTan are meant for complex and real usage, not a mixture. However, it is disappointing that the simplification does not work.