### 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)

delay_on_limit_secs=1.0,
        verbose=False,
        conf=None,
        rerun_failed_tasks=False,
        run_backwards=False,
        run_at_least_once=False,
        continue_on_failures=False,
        disable_retry=False,
    ):
        """
        Run the DAG.
        :param start_date: the start date of the range to run
        :param end_date: the end date of the range to run
        :param mark_success: True to mark jobs as succeeded without running them
        :param local: True to run the tasks using the LocalExecutor
        :param executor: The executor instance to run the tasks
        :param donot_pickle: True to avoid pickling DAG object and send to workers
        :param ignore_task_deps: True to skip upstream tasks
        :param ignore_first_depends_on_past: True to ignore depends_on_past
            dependencies for the first set of tasks only
        :param pool: Resource pool to use
        :param delay_on_limit_secs: Time in seconds to wait before next attempt to run
            dag run when max_active_runs limit has been reached
        :param verbose: Make logging output more verbose
        :param conf: user defined dictionary passed from CLI
        :param rerun_failed_tasks:
        :param run_backwards:
        :param run_at_least_once: If true, always run the DAG at least once even
            if no logical run exists within the time range.
        """