Can he open the trading account with Zerodha without opening any other account with HDFC, Axis or Yes bank?
Yes, you can open multiple demat accounts with other brokers.
No, you cannot open an account at Zerodha by mapping your NRE-PIS account with ICICI Bank.
Zerodha hasn’t yet started offering PIS services through ICICI. You will have to come through either one of these three banks banks only - HDFC, Axis or Yes bank. These are the 3 banks with which Zerodha offers PIS services.
An NRI cannot have multiple Demat accounts. You can have one NRE and NRO Demat account.