On the real line, I understand that $x^{-1}dx$ is a measure invariant by scaling, where $dx$ is the usual measure.
However, for the unitary group, for example, I have seen Haar measure written analogously as $U^{-1}dU$. But what does the symbol $dU$ stand for, in this expression? Is $dU$ unitary? Is $U+dU$ unitary? Does it mean that I can use any parametrization whatsoever and implement small changes in the parameters?
You (or/and your sources) are confusing the Haar measure and the Maurer–Cartan form on Lie groups, which for matrix groups indeed can be written as $g^{-1}dg$. Here $g$ is understood as a smooth map (local parameterization) to the Lie group and $dg$ is the differential of this map.