Yes, anyone can take that key. And that's by design: the anon key (newer projects call it the publishable key) is meant to be public, like a building's street address. What decides who gets in are the locks: Row Level Security.
So the real question is: is RLS enabled on every table, with sensible policies?
Check:
- Dashboard → Table Editor: any table marked 'RLS disabled' / 'Unrestricted' is readable and writable by anyone with your public key. Fix those first.
- Dashboard → Advisors → Security Advisor: lists tables without RLS and other issues.
- Test: log in as user A, try to read user B's rows from the console. Should return nothing.
The key that must never be in the browser is the service_role / secret key. That one bypasses RLS.