### 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)

"""Check ``schedule_interval`` and ``timetable`` match.
        This is done as a part of the DAG validation done before it's bagged, to
        guard against the DAG's ``timetable`` (or ``schedule_interval``) from
        being changed after it's created, e.g.
        .. code-block:: python
            dag1 = DAG("d1", timetable=MyTimetable())
            dag1.schedule_interval = "@once"
            dag2 = DAG("d2", schedule="@once")
            dag2.timetable = MyTimetable()
        Validation is done by creating a timetable and check its summary matches
        ``schedule_interval``. The logic is not bullet-proof, especially if a
        custom timetable does not provide a useful ``summary``. But this is the
        best we can do.
        """
        if self.schedule_interval == self.timetable.summary:
            return True
        try:
            timetable = create_timetable(self.schedule_interval, self.timezone)
        except ValueError:
            return False
        return timetable.summary == self.timetable.summary
    def validate(self):