### 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)

"""tzinfo derived concrete class named "+0530" with offset of 19800"""
            
            _offset = datetime.timedelta(seconds=19800)
            _dst = datetime.timedelta(0)
            _name = "+0530"
            def utcoffset(self, dt):
                return self.__class__._offset
            def dst(self, dt):
                return self.__class__._dst
            def tzname(self, dt):
                return self.__class__._name
        start = datetime.datetime(2018, 3, 25, 10, tzinfo=UTC0530())
        dag = DAG("tz_dag", start_date=start, schedule="@hourly")
        when = dag.following_schedule(start)
        assert when.isoformat() == "2018-03-25T05:30:00+00:00"
    def test_dagtag_repr(self):
        clear_db_dags()
        dag = DAG("dag-test-dagtag", start_date=DEFAULT_DATE, tags=["tag-1", "tag-2"])
        dag.sync_to_db()
        with create_session() as session:
            assert {"tag-1", "tag-2"} == {
                repr(t) for t in session.query(DagTag).filter(DagTag.dag_id == "dag-test-dagtag").all()
            }
    def test_bulk_write_to_db(self):