To monitor risk of a client portfolio, does it make sense to accumulate Greeks across different underlying? If yes, how can Greeks be normalized across different underlying?
It depends on what kind of 'Greek' and 'underlying' you are referring to.
Aggregating the sensivity of various equity positions (e.g. options, futures) to interest rates could for instance make sense if you want to evaluate the sensitivity to that common risk factor.
On the other hand, aggregation of underlying-dependent quantities (such as Delta) across distinct underlyings, does not make sense, at least in the equity world.
That being said, aggregation across a different positions/instruments written on a given underlying asset does of course make sense (e.g. macro hedging of the Delta, or bucketed Vegas).