Does my registered name with Bagh have to be the same as the name on my identification document and account(s) used in depositing/withdrawing?

Yes. You must register your Bagh account with your legal name and identity, which must be consistent with your Bkash/Rocket/Nagad/OK wallet/SureCash/tap/upay account(s), bank account(s), or other payment accounts used to deposit or receive funds in your Bagh account.