Bank Mode should also be integrated with Mozilla Firefox, chrome and Microsoft Edge.
Bank Mode is a feature available only in Avast Secure Browser.
If Avast were to make this feature available in other browsers, they would have to work with other browser companies to implement their Bank Mode feature.
You are correct in that Bank Mode feature only works with Avast Secure Browser. Avast Secure Browser is owned and maintained by Avast.
See attached below for examples of Bank Mode not in action and in action: