Milk teeth are baby teeth. In other words, she hasn’t even lost her baby teeth yet and she thinks she’s “got herself together” (しっかりしている) moreso than her older sister.
This is right after her sister says something along the lines of “can’t you just make up with (them)?” (We don’t know who she’s talking about). To which her older sister sharply (ぴしゃり) replies “It’s an adult problem!”. In other words, stay out of it kid, you’re too young to understand.
I’ve neither read the English version nor seen the movie, but based on this little section, there appears to be some sort of issue with the mayoral election (or someone involved with it). Grandma angrily pulled the plug on the speaker as soon as it got brought up, and then Mitsuha derides the idea of the election again right after snapping at her sister in the passage you quoted. Or maybe I’m overthinking it.
Edit: Reading a little further tonight, the election stuff all makes sense.
Edit #2: Found some nice info on かんか
It’s a command form in Kansai-ben from what I can see. According to a post I found on hinative:
In the Kansai dialect, ___んか (or かい) expresses an order,
はよ食べんかい (eat quickly! Kansai)
早く食べなさい (eat quickly! Standard)
So 胸(を)張って歩かんか(い)! = 胸を張って歩きなさい！(Walk with pride!) <— someone else can probably translate that better