[FEATURE]: Show which database/connection Studio is currently connected to

### Feature hasn't been suggested before.

- [x] I have verified this feature I'm about to request hasn't been suggested before.

### Describe the enhancement you want to request

Right now, once Studio (`drizzle-kit studio`) is open at `local.drizzle.studio`, there is no visible indication of *which* database it's actually talking to. no database/host name in the header, no browser tab title, nothing in the page itself. The only way to check is to manually run `select current_database();` (Postgres) in the SQL runner every single time.

This becomes a real data-safety risk once you routinely point Studio at more than one database — e.g. a local dev instance, a shared dev DB, a dedicated test DB, or even prod DB. It's easy to forget to check and end up editing/deleting rows in the wrong database, especially since the UI looks identical regardless of target.

Concretely, I'd like one of:

- Studio's header/sidebar (or document.title) always shows the resolved connection target (host + database name, or a **custom label**), so it's visible at a glance without running a query.
- `drizzle.config.ts` (or the `studio` CLI) gains an optional `name`/`label` field that gets rendered in the UI, so visually distinct configs/environments are easy to tell apart.

Happy to clarify further or provide more context if needed.