Of-course you can trade intraday in BSE as well as NSE with other brokers. May be the sales person is not well informed or ICICI has this weird rule.
However, do note that some stocks cannot be traded in intraday for regulatory reasons on BSE as well a NSE. These are from BSE's T segment(and mat be S segment also) and equivalent to NSE's BE series. Google for more info on this.