IIRC Christians prohibited charging interest on loans (usury), not banking per se. I suppose a bank could be operated by Christians that simply accepted deposits and charged a sum for safekeeping with no loans whatsoever. I don't know that anything like that ever happened in practice, however, since anybody with enough money to require safekeeping in a vault probably had their own vault.

Likewise an interest free loan would be permissible, though there wouldn't be great incentives to offer them except to family members, for example.

What happened in the Islamic world was partnerships instead. If a merchant wanted to make a trade expedition funded by a banker, they'd set up the expedition as a partnership with the banker providing the funds and the merchant doing the work, splitting the profits, in a way that's economically equivalent to lending money at interest. Similarly, insurance syndicates tended to be organised as a bunch of merchants getting together and agreeing to pool a certain amount of funds that would be used to repay any of their losses.

(It works up to a point, but it's hard to scale. You can emulate a loan contract between two parties, but you can't have a bond market because the "loans" aren't tradeable)