Hi @Yoni Wolovitz ,
I understand that you are trying to grant Power BI access to the external users, but these users need to log in with a different username from their organizational domain. Based on the Power BI External Users guide, it appears that you have fulfilled the prerequisites for setting this up. But if you invited the users with the "email@example.com" email, Azure will be expecting that address rather than the other one.
Since Azure is expecting the domain email, you have the option to enable sign-in to Azure AD with email as an alternate login ID so that they can use their non-UPN email.
Another thing you could do instead of changing the email address is add the alias for the user, so you would add firstname.lastname@example.org as an alias to name.surname@org .com. https://learn.microsoft.com/en-us/microsoft-365/admin/email/add-another-email-alias-for-a-user?view=o365-worldwide
Otherwise the invited emails need to be added as guest accounts in the tenant, and you will need to re-invite the users.
Let me know if this helps.
See also: How to update b2b user upn/mail