Hi Matteo,
My system is pretty much what you mention in your example, a plain Moodle installation (3.6 after the upgrade coming from 3.2) with external dabatase enrolment and external database authentication, and just a couple fo 3rd party plugins.
What happened at first was that I had only changed the external database enrolment dbtype setting to mysqli, leaving the external database authentication auth_db setting to mysql since that wasn't mentioned in the post I linked and I didn't think of it.
That was giving the errors I mentioned, so I started from scratch (reverted my installation to Moodle 3.2 again from a backup). This time I did change both settings to mysqli (enrolment and authentication) before doing the upgrade, but still got the error. I decided to check for both values in the database and they still showed as mysql, so I corrected them manually and everything worked.
It does sound like a weird issue but oh well, at least it's fixed now