Zebpay is an exchange where you can buy/sell or trade your bitcoins, it is not generally a wallet but an exchange wallet.
Bitcoin core is completely different here, it is a wallet in which you need to download the whole blockchain which covers all the transactions since the start till date and lets you send/receive bitcoins to/from anyone, zebpay has a wallet too from which you can do the same thing, but it is an exchange too and such utilities are not provided in Bitcoin Core.