@Riccardo Bassi - Thanks for the question and using MS Q&A platform.
It seems like you are having trouble applying sensitivity labels to documents in SharePoint. There are a few conditions that need to be met in order to apply sensitivity labels to documents in SharePoint.
Firstly, the sensitivity label scope must be configured for Groups & Sites. Secondly, the label must be available to Azure AD. Thirdly, the current signed-in user must have sufficient privileges to assign labels. The user must be either a Global Administrator, Group Administrator, or the group owner. Lastly, the current signed-in user must be within the scope of the sensitivity label publishing policy.
If all of these conditions are met, you should be able to apply sensitivity labels to documents in SharePoint. If you are still unable to apply sensitivity labels, it is possible that there may be an issue with the configuration of your sensitivity labels or policies.
Regarding the label that should have been applied by default according to the policies, it is possible that the label is not published in the Microsoft Purview compliance portal or it is not available to the user that is signed-in. Please check with your administrator for more information on how to get access to the label.
I hope this helps! Let me know if you have any further questions.