Banks do two things:
1. Store large amounts of money securely.
2. Transfer money.
Bitcoin does both.
Bitcoin 100% does not do #1. You personally are the guy in charge of security of your bitcoins, just like you are in charge of your own regular wallet
#2. In a way. Bitcoin proposes the protocol for exchanging "money" (bitcoins) or "value" or whatever you want to call it. Again, this is slightly different from your point. Actual clients (or implementations) of the bitcoin protocol are a means of transferring the bitcoins
Cheers
Rob