Yes,it is very necessary to have a Bitcoin wallet because it is the only storage unit for Bitcoin and without having one, one can't store his or her Bitcoin since it a digital money.
Without a Bitcoin wallet, we can't send or receive Bitcoin, Bitcoin isn't like physical money that we can leave in our pocket, so as virtual money we also need virtual wallet to transacts any business with.
Bitcoin wallet help our Bitcoin to be safe and secured.Leaving our money on different Bitcoin earning site isn't safe because sites can go down any time and our hard earn Bitcoin would be gone with the site.
Bitcoin wallet helps us send money anonymously. The wallet just generate some code for us to use as our address, no need using ones actual name.