Should I relate BETA of a stock to Nifty 50 or it's sectoral index?

While understanding beta of SBI(1.93), does it mean it outperforms/underperforms Nifty 50 or Nifty PSU bank by 93% when index goes up/down?

Beta is usually measured keeping the Index in perspective, so in this case, it would be the Nifty 50 index. You can calculate the beta wrt to Bank Nifty as well, section 11.6 in this chapter helps you understand how this can be done.