Neither $PFLAA or $PFLAU Flarm sentences have the aircraft callsign. However, both have the ID which is a unique 6-digit hex identifier (either the aircraft's ICAO code, the default equipment ID or what someone has set). Flarm and SoftRF displays use a table file from FlarmNet or OGN to look up the aircraft registration based on the ID. See Flarm Technical Document FTD-012 which includes following examples:
$PFLAU,2,1,2,1,1,-90,2,50,75,1A304C* FLARM is working properly and receives two other aircraft. The most dangerous
of these aircraft has the ID “1A304C” and it is at 9 o’clock, position 50m above
and 75m away. Level 1 alarm.
$PFLAA,0,-1234,1234,220,2,DD8F12,180,,30,-1.4,1*
There is a glider in the south-east direction, 1.7km away (1.2km south, 1.2km
east), 220m higher flying on south track with a ground speed of 30m/s in a slight
left turn with 4.5°/s turning rate, sinking with 1.4m/s. Its ID is a static FLARM-ID
“DD8F12”. There is no danger.