### 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)

"""
        Return a list of dag run execution dates currently running.
        :return: List of execution dates
        """
        runs = DagRun.find(dag_id=self.dag_id, state=DagRunState.RUNNING)
        active_dates = []
        for run in runs:
            active_dates.append(run.execution_date)
        return active_dates
    @provide_session
    def get_num_active_runs(self, external_trigger=None, only_running=True, session=NEW_SESSION):
        """
        Return the number of active "running" dag runs.
        :param external_trigger: True for externally triggered active dag runs
        :param session:
        :return: number greater than 0 for active dag runs
        """
        query = select(func.count()).where(DagRun.dag_id == self.dag_id)
        if only_running:
            query = query.where(DagRun.state == DagRunState.RUNNING)
        else:
            query = query.where(DagRun.state.in_({DagRunState.RUNNING, DagRunState.QUEUED}))
        if external_trigger is not None:
            query = query.where(
                DagRun.external_trigger == (expression.true() if external_trigger else expression.false())
            )
        return session.scalar(query)
    @provide_session
    def get_dagrun(