### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

d = {}
        d["is_picklable"] = True
        try:
            dttm = timezone.utcnow()
            pickled = pickle.dumps(self)
            d["pickle_len"] = len(pickled)
            d["pickling_duration"] = str(timezone.utcnow() - dttm)
        except Exception as e:
            self.log.debug(e)
            d["is_picklable"] = False
            d["stacktrace"] = traceback.format_exc()
        return d
    @provide_session
    def pickle(self, session=NEW_SESSION) -> DagPickle:
        dag = session.scalar(select(DagModel).where(DagModel.dag_id == self.dag_id).limit(1))
        dp = None
        if dag and dag.pickle_id:
            dp = session.scalar(select(DagPickle).where(DagPickle.id == dag.pickle_id).limit(1))
        if not dp or dp.pickle != self:
            dp = DagPickle(dag=self)
            session.add(dp)
            self.last_pickled = timezone.utcnow()
            session.commit()
            self.pickle_id = dp.id
        return dp
    def tree_view(self) -> None: