The services are not related the airplane you are flying, but are a combination of the airport and the parking code (as defined in the scenery AFCAD) of the selected gate.
The assumption is that, you can't expect to be serviced by your preferred handling company *everywhere* and, since FSX allows to fly any airline anywhere, it's more realistic to link services to the airport, if I fly an Alitalia airplane to JFK or KLAX, I can't expect to find the same handlers as in Italy.
Since GSX ALSO use the airline parking code of the gate (if the scenery supports it, of course), you can indicate some kind of preferred services, by parking your airplane on a gate that has your airline code, which is more realistic too even without considering handlers.
So, if the scenery you use has its parkings correctly set with the codes of your airline AND your preferred handling company is available at that airport, chances are you'll get your preferred handler.
If not, you should edit the scenery AFCAD to add your airline code to the gates your airline normally use, and go park there.
If that airport is not serviceable by that GSX handler (Swissport in your example) you should add its ICAO to the SIM.CFG of the GSX vehicles in the Swissport livery.
To know more about this, download the Paintkit, the documentation covers this matters in great detail.