I finally figured it out. The main issue was that I initially linked my new External ID tenant to an existing subscription that was still associated with my home directory, which caused problems.

To resolve it, I created a new subscription and made sure to assign it directly to the new tenant / directory.

After that, I was able to switch directories again — and this time, MFA worked as expected, and I successfully switched tenants.

Additionally, I now see that I’m assigned the Global Administrator role by default in the new tenant, just as expected and as confirmed in the Microsoft Docs

By default, the user who creates a Microsoft Entra tenant is automatically assigned the Global Administrator role.