That Hydreigon starred in a PokéStar Studios movie and got a strange ending. When you get a strange ending with one of your Pokémon, they get that animation.
I can confirm this answer. I have a Leavanny that participated in a movie and received a Strange Ending. What it does is basically reminds the player using them that they are movie stars, have lots of fans, and will generate more movie revenue as a result if you continue using that Pokémon without a bad ending in future movies.
If you want to remove the animation from your own star Pokémon (who would?), trigger a Bad Ending.
If you want your own Strange Ending, here's a simple one to try.
Brycen-Man
"The {Insert Pokémon} Girl/Kid vs. Brycen Man"
PLOT: A hero battles the Masked Man, who is targeting the amusement park.
MISSION: Knock out an opponent every scene.
STRANGE ENDING: Miss the first move, but do not get KO'd.
SCENES: 2
Brycen / Masked Man Brycen-Man:
Pawniard (Lv. 15 Dark/Steel)
Ability: Defiant
Brick Break/Aerial Ace
Vullaby (Lv. 13 Dark/Flying)
Ability: Big Pecks
Faint Attack/Air Slash
In other words, use a Lv.20 or higher or pretty much any type, or one that is type resistant at a similar level and won't get knocked out. If you get KO'ed, you trigger the BAD ending, so make sure you have someone who can survive at least the first turn (knocking out the opponent the second turn, but leaving the second Pokémon unused; perhaps even survive two turns without attacking). In fact, if you miss the first turn, there might not even BE a second turn.
Make sure you choose the "good" line, which is "Bring it on!"; I don't know if choosing the Bad line changes to a Bad ending, but why take the chance?
After completing this, you will have a star over your Pokémon's head. :)
Cyclone