When flying LIS-MXP-FCO with an AZ ticket, but the first leg being operated by TP (on a code-share), I was denied access at the AF owned lounge at LIS (but then let in by courtesy) and at the AZ owned MXP lounge. I complained but after several exchanges with FB, they clarified that I would have no access indeed.
In my opinion, you should have had access in MXP, based on the fact that you were connecting from an international Skyteam flight, since you were on the AZ coded flight from LIS.
However, if your boarding pass only showed the TP information, they could have estimated it was not a Skyteam flight.
Edit: just checked the Flying Blue rules, and they were correct:
* When travelling on or connecting to/from an international SkyTeam-operated flight on the same day.
Your international flight was not Skyteam operated, thus no access.
Cheers,
GenevaFlyer