GA pilot reported a NMAC in the traffic pattern when another aircraft was on downwind at the same time. The other aircraft took evasive action.

Narrative

We were cleared for takeoff on Runway 6 and instructed to make a right downwind departure for VFR flight to the west. An L-39 using a callsign was on an approximately 5 mile final. Additional traffic was a flight of 2 led by an Extra that was having a back and forth with ATC from the time we checked in on Tower frequency. We departed and turned to the right downwind. ATC was still trying to vector the Extra and the L-39 asked tower to change to a low approach. We watched the L-39 do a low approach but did not hear Towers instructions for them on the go. After ~30 seconds we got an auditory Traffic advisory from ADS-B showing 100 ft. low and directly behind us. Looking back I saw nothing. The traffic advisory then showed coaltitude and on top of us; so I looked again and saw the L-39 passing on the left co-altitude at approximately 100 ft away (as ID'd by approximate military formation sight references). They seemed to notice us approximately line abreast as evidenced by a wing rock and taking spacing. They then climbed and once no factor crossed our flight path to continue on their direction. Tower was still involved in trying to direct the Extra flight at the time and eventually gave them a number. As further evidence of Tower's task saturation due to the Extra flight; we were never given a frequency handoff. We continued on with an otherwise uneventful flight. On the ground; I called the Tower to ask what had happened and they said the Extra flight was the issue they were working and none of us had done anything wrong; I am however still confused on how both we and the L-39 both ended up on the same downwind.