If a potential sugar daddy asks you to send money to him, just dont! Youre supposed to be the recipient of monetary gifts and s/hes meant to be the provider: those roles should never, ever be reversed. Dont share your bank account, credit card or debit card numbers (they can all be used to steal money from you).

Should I tell my sugar daddy who I bank with?

No. He doesnt need that information to send you money. Banks can send Pop Money, which is basically an electronic check / funds transfer to you. You should call your bank or his bank to look into alternate transaction methods.