The two use cases you propose above are permitted by AGPL.
The license refers more to the code itself than it’s usage. GPL permits you to use the application as is wherever you want, but IF you modify the code, AND someone asks you for those modifications you MUST share them
If you copy the code and put it inside your application, the WHOLE application becomes GPL and you MUST share ALL of it if someone asks for it.
So if you are using Metabase as you download it, with the branding and the native embedding support as you describe above I’d say you are good.
Note that I am not a lawyer, but I have been reading about these issues for many years now.