How do civil aircraft flying in formation handle ATC?
As long as the aircraft are in formation, they are controlled as a single aircraft.
From US DoT OrderJO 7110.65T ATC General Control:

2-1-13. FORMATION FLIGHTS
a. Control formation flights as a single aircraft. When individual control is requested, issue advisory information which will assist the pilots in attaining separation. When pilot reports indicate separation has been established, issue control instructions as required.

The decision to break the flight lies with the pilot in command, and the ATC then controls the aircraft individually.


Separation responsibility between aircraft within the formation during transition to individual control rests with the pilots concerned until standard separation has been attained.

Upon break-up of the formation flight, the controller initiating the break-up shall ensure that all aircraft or flights are assigned their proper equipment suffix.