Imagine this. Ash and Go go to Hoenn next episode
Sakuragi: Ash, Go, Wait. I need to give you this
Go: What is this, Professor
Ash opens the bag: CANDIESSSSS. Yummy.
Sakuragi: Those are not for you, but for your pokemon. Maybe they recognize the smell and taste.
That they have a connection, doesn't mean Mr Mime belongs to her. It can come and go as it pleases, with the main difference that it doesn't want to leave (Ash and Meloetta have a connection, but she still doesn't belong to him).
Ash catching Mr. Mime, doesn't change all that. Maybe he is...
Dragonite: Lance, Iris, Team Rocket
Sobble and Grookey: Most likely gonna be Ash's, seeing the are both next to Pikachu
Mimey: Will Ash catch it. I hope so. Would be his first Fairy and Psychic type.