An update on the current situation on my phone.
Android 9 (pie) on Motorola One, August security updates applied
Brave version: 1.3.2, Chromium 76.0.3809.132
Chrome version: 77.0.3865.73
Password manager: Bitwarden version 2.2.2 (2072)
Experience in Brave: Using the Amazon.co.uk login page I sometimes get the Bitwarden pop up when I put the cursor into a login field. While writing this, I tested and logged in successfully. However, I then logged out and tried again, and the pop-up no longer appeared.
Experience in Chrome: The same.
Closing Brave and Chrome from the Android app switcher, then starting them again seems to revive the pop-up but just for the first time. Subsequent attempts to login have the same experience.
So it looks to me that Brave might indeed have the API enabled, but something else is preventing the pop-up.
If you are having similar challenges with another password manager, it would be great to hear more.