According to the ICAO phraseology, AIM, and pilot/controller glossary, Pan-Pan, Pan-Pan, Pan-Pan is a prefix used to denote an urgency message. I have seen no source that states that one should add pan-pan to one's callsign, even for non-urgency messages. Arguably the initial call about flaps not working should have been a pan-pan. The subsequent messages about holding fixes and direct VORs did not need a pan. Here's another example straight out of the relevant ICAO guide, example of Emergency Communications:
RTF Emergency Communications
MAYDAY, MAYDAY, MAYDAY, Metro Control, Big Jet 345, main electric failure,
request immediate landing at Metro, position 35 miles north west of Metro,
heading 120 flight level 80 descending, 150 persons on board, endurance three
Big Jet 345, Roger the MAYDAY, turn left heading 090, radar vectors ILS runway
Big Jet 345 request runway 09
Big Jet 345, roger, turn right heading 140 for radar vectoring runway 09,
descend to 3000 feet, QNH 995, report established
Big Jet 345, heading 140, descend to 3000 feet QNH 995 , report established
localiser runway 09
As you can see, Mayday Mayday does not become part of the call sign. It is used for the initial call-up to denote a distress call or distress situation, but each message after that is not itself a distress call. Same with Pan pan.
Likewise, the AIM states that the initial communication should be prefixed with Mayday x 3 or Pan-Pan x 3. It does not state that the mayday or pan-pan becomes part of the callsign.
Not a big deal, but the AF pilot just sounds kind of ridiculous. Every time he spoke I rolled my eyes. But the French have that effect on me in general.