My reputation score of parent and meta site is automatically synchronized after some time. But badges are counted differently for both site. Why?
Reputation is always inherited from the parent site.
You cannot gain or lose reputation(*) on the per-site meta.
This also means that some reputation related functions like the rep graph and bounties are not enabled on the per-site meta.
*: however, there is one exception:
extreme misbehavior on the meta site will affect your parent site reputation. And not in, shall we say, the “good” way.
Voting up or down does not affect reputation.
You are now free to vote purely based on post content, without worrying about how your vote might positively or negatively affect someone’s reputation score.
So there is no actual reputation on Meta, and you start with no badge either.
Except that the badges will increase, as per your actions on Meta.
Your "Meta reputation" will not (since it is a mere reflect on your "Parent reputation" (i.e. "your reputation on the parent site").