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The only use for your carta di soggiorno di familiare di un cittadino dell'unione in the Philippines airport is as documentation that you are authorized to fly to Italy. This is required on your return journey.
Upon arrival in the Philippines, you won't have any use for the card. You will need a passport to enter the Philippines.
You will need to show the card to the airline when you board your flight to return home to Italy. This is not a choice. Otherwise, the airline will not permit you to board the flight, because without that card you lack adequate documentation to enter the Schengen area.