I think Fidelity does allow this, but I haven't bothered with it since I use a password manager.
Fidelity has a brokerage account, free checks, free ATM withdrawals via debit card, maybe also your 401k, free money wires, automatic investment etc.
The only thing they don't have are branches where you can deposit cash, but that's really never necessary - in an extreme case you can open another bank account, deposit cash, transfer to fidelity and immediately close it.
I'm not sure why anyone uses a bank other than Fidelity.
Fidelity does it through either SMS or Symantec’s Validation and ID Protection (VIP) Access app. I called and asked if they support another app and they said they don't. Why they couldn't use another (read: non-Symantec) 2FA is beyond me.
Fidelity has a brokerage account, free checks, free ATM withdrawals via debit card, maybe also your 401k, free money wires, automatic investment etc.
The only thing they don't have are branches where you can deposit cash, but that's really never necessary - in an extreme case you can open another bank account, deposit cash, transfer to fidelity and immediately close it.
I'm not sure why anyone uses a bank other than Fidelity.