Well, in my opinion, banks will only accept people to use their money to buy Bitcoin. With this strategy, banks may know the buyer identity and also earn some profit with the fee they gain from selling Bitcoin. However, no matter how hard they try, banks will never accept bitcoin as a main currency because there is a big discrepancy between banks and Bitcoin
Banks aren't dumb, neither are the governments that put up all kinds of rules these banks abide by. They are well aware of the fact that there are plenty of mixers a.k.a. tumblers, where people can simply get rid of their "marked" coins, and get "clean" coins in return. Of course they can investigate the entire transaction log within the blockchain in an attempt to trace these "cleaned" coins, but there are zero guarantees that you'll ever find them. Especially with the features that mixers offer, where you can split a certain amount into 5-10-20 or even more smaller pieces, each being sent to a different address. If you so wish you can decide to run even these coins through a mixer if you're really paranoid. Nothing stops you then from using the coins that you get back in the end for whatever illegal purpose, or to finance terrorism. Governments won't ever allow banks to sell Bitcoins.