The double strike ability granted by Atarka is much more akin to an aura or an equipment (although it lasts only until end of turn), rather than a static ability of the card itself.
This keeps all the following scenarios lined up:
1) You have Atarka and
Shivan Dragon.
Shivan dragon attacks, gaining double strike.
Shivan Dragon becomes something else, say an opponent casts
Turn to Frog on it. The Frog
Shivan Dragon still has double strike.
2) Scion has
Fireshrieker attached to it (granting double strike). Scion becomes a copy of another dragon. Scion doesn't leave the battlefield, it just becomes a copy. That dragon will still have
Fireshrieker equipped and will still have double strike.
3) You have Scion and Atarka. Scion swings, gaining double strike until end of turn from Atarka. Scion becomes a copy of any other dragon. It will still have double strike from the Atarka trigger.
Happy Brewing!