To upgrade the SSRS edition you just need to re-run the SSRS installer and choose "Upgrade Edition". There you can enter your license and upgrade the edition.

To upgrade to a newer version of SQL AND upgrade to a new edition you'll need to run the setup twice:

1. Upgrade SQL to version and edition you want (e.g. 2019 Express to 2022 Standard).
2. Download and run the SSRS installer that matches your new version of SQL to upgrade the SSRS version to match the SQL server.
3. Run the SSRS installer again to upgrade the edition to match the SQL server.

I just dealt with this issue today; the whole upgrade process was a bit frustrating and not nearly as streamlined or seamless as the MS Documentation implied.