### 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 BaseOperator(task_id=task_id).as_teardown()
        def teardown_task_f(task_id):
            return BaseOperator(task_id=task_id).as_teardown(on_failure_fail_dagrun=True)
        def work_task(task_id):
            return BaseOperator(task_id=task_id)
        def setup_task(task_id):
            return BaseOperator(task_id=task_id).as_setup()
        def make_task(task_id):
            """
            Task factory helper.
            Will give a setup, teardown, work, or teardown-with-dagrun-failure task depending on input.
            """
            if task_id.startswith("s"):
                factory = setup_task
            elif task_id.startswith("w"):
                factory = work_task
            elif task_id.startswith("tf"):
                factory = teardown_task_f
            elif task_id.startswith("t"):
                factory = teardown_task
            else:
                raise ValueError("unexpected")
            return dag.task_dict.get(task_id) or factory(task_id=task_id)
        return (make_task(x) for x in input_str.split(", "))
    @staticmethod
    def cleared_downstream(task):