BSE is the oldest exchange in India and was started off by a group of brokerage houses, these guys ran the show for quite some time and had no real competition, at that time the trading used to happen using the open outcry method, this was having inherent risks, like a risk of bad/short delivery, frauds, losses of share certificates etc.
Then came NSE,they started off with Screen based trading system, essentially where brokerages could send orders to the exchange electronically rather than, the orders going through people, this also lead to dematerialization and introduction of shares being held in the demat account rather than in the physical Certificate form.
Also NSE started allowing trading in derivatives, which BSE thought wouldn’t be successful/popular, unfortunately against their expectations, traders here accepted derivatives, BASE introduced derivatives much later, by then NSE was way ahead.
This is the reason why volumes are more on NSE than on BSE.
In India, there are two depositories, NSDL( backed by NSE) and CDSL( backed by NSE) these are two independent companies and have nothing to do with these exchanges, so essentially if you open an account in a brokerage house like Zerodha, you will be allowed to buy/ sell securities on NSE and BSE, now when you buy shares, these will go to your demat account, which could be with the Depository that the DP has tied up with(CDSL in case of Zerodha Securities Pvt Ltd)
Buying shares to me is like buying veggies, you may buy it from the footpath vendor (BSE) or the supermarket (NSE) once you have to store them, you do put them in the same refrigerator ( your demat account- with CDSL or NSDL)