With version 12 I get a full, if complicated, solution:
In[1]:= Solve[{x == r*Cos[theta], y == r*Sin[theta]}, {r, theta}]
Out[1]= {{r -> -(x^2/Sqrt[x^2 + y^2]) - y^2/Sqrt[x^2 + y^2],
theta -> ConditionalExpression[
ArcTan[-(x/Sqrt[x^2 + y^2]), -(y/Sqrt[x^2 + y^2])] + 2 \[Pi] C[1],
C[1] \[Element] Integers]}, {r ->
x^2/Sqrt[x^2 + y^2] + y^2/Sqrt[x^2 + y^2],
theta -> ConditionalExpression[
ArcTan[x/Sqrt[x^2 + y^2], y/Sqrt[x^2 + y^2]] + 2 \[Pi] C[1],
C[1] \[Element] Integers]}}
