Well before going to measurement I think it’s appropriate to start by definition. The term shadow bank is not always 100% consistently applied but most often the a shadow bank is defined as follows:
Shadow banks are financial intermediaries that conduct maturity, credit, and liquidity transformation without explicit access to central bank liquidity or public sector credit guarantees.
This is how you see shadow banks defined a lot in research (and also other definitions come close). Moreover, this is definition used in research coming from Fed such as this paper from NY Fed or this paper from St Luis Fed for example.
To my best knowledge there is no unified measure/indicator that’s being recorded (like let’s say GDP/inflation). Rather like this literature review from NY Fed mentions people take the definition of shadow bank and then calculate the proportion of for example financial liabilities hold by shadow banks relative to ordinary banks or other types of financial institutions. Maturity and credit transformation calculated as proportion per type of bank is also used. In wider research I have seen other measures as well.