### 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)

"""
        Test if the schedule_interval will be auto aligned with the start_date
        such that if the start_date coincides with the schedule the first
        execution_date will be start_date, otherwise it will be start_date +
        interval.
        """
        dag = DAG(
            dag_id="test_scheduler_auto_align_1",
            start_date=timezone.datetime(2016, 1, 1, 10, 10, 0),
            schedule="4 5 * * *",
        )
        EmptyOperator(task_id="dummy", dag=dag, owner="airflow")
        next_info = dag.next_dagrun_info(None)
        assert next_info and next_info.logical_date == timezone.datetime(2016, 1, 2, 5, 4)
        dag = DAG(
            dag_id="test_scheduler_auto_align_2",
            start_date=timezone.datetime(2016, 1, 1, 10, 10, 0),
            schedule="10 10 * * *",
        )
        EmptyOperator(task_id="dummy", dag=dag, owner="airflow")
        next_info = dag.next_dagrun_info(None)
        assert next_info and next_info.logical_date == timezone.datetime(2016, 1, 1, 10, 10)
    def test_next_dagrun_after_not_for_subdags(self):
        """
        Test the subdags are never marked to have dagruns created, as they are
        handled by the SubDagOperator, not the scheduler
        """
        def subdag(parent_dag_name, child_dag_name, args):