Well, my question is pretty simple: when archaon flips and pins 2 enemy disks, does his ability activate for each disk, I mean, would each disk suffer 2 magic damage? Or any ability that says "when you engage" applies only once per flip, making both pinned disks suffer just 1 magic damage (plus impact)?

Both disks would suffer 2 physical damage from his impact, then his ability would trigger, dealing 1 magic damage to each enemy disk within short range. So both enemy disks that he pinned would take a total of 3 damage. Since the engagement is simultaneous, his ability would only trigger once.

Actually, the ability triggers more than once. In this instance, it would trigger twice, dealing two arcane damage.

Per the designer's words : (and courtesy of the Sniper Troll's rules compendium)

Q: If a disk has an ability that says “when this disk engages an enemy” (i.e. Archaon), does its ability trigger multiple times when it engages more than one enemy at once?
A: Yes, Archaon and similar abilities trigger each time they engage a disk. This is different than an ability that says “When this disk engages at least 1 enemy…” which would only trigger once.