Can a third party, such as a bank, distribute Google Chrome extensions to only its customers?

With a little jiggery-pokery, distribution of private (non store) extensions is possible. However, it requires a little know-how on behalf of the customer, and it's not technically supported by Google. More complex methods involve registry editing, but you can do it by downloading and dragging installation files to the extension management page. It's not ideal in any way, and probably too much hassle.

As Chrome is open source (you can edit and redistribute the entire program), you could possibly look at developing the extension and distributing a bank-branded version of the Chrome software with the extension as part of the software. It's less likely to be taken up by customers, as they're switching from a well-known browser, but it's a more elegant solution than getting them to install an extension via backdoor routes.

I hope this helps.