You are not necessarily doing anything wrong — this is a common limit with filling into other apps.
A few things worth checking:
- Only one autofill service can be active at a time on your phone, so check in your phone settings that the browser is actually set as the default autofill service, and that no other service is also trying to fill the same field.
- Some fields, especially in security-sensitive apps, deliberately block filling or are not recognised as password fields, so the prompt can appear and still put nothing in. That would explain why it works with copy and paste but not with filling.
- Try tapping the field first, waiting a second for the prompt, and then choosing the entry, rather than tapping straight into it. If that still puts nothing in, try the long-press fill option if your phone offers one.
- For the safe itself, unlocking it from inside the password manager app is usually the more reliable route, since its own unlock screen is not meant to be filled from outside.
It would help to know which phone and system version you are on, whether the browser is set as your default autofill service, and whether filling works in an ordinary app, such as a notes app, so it is easier to tell whether this is a setting problem or a blocked field.