### 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)

``Environment`` for template rendering
        **Example**: to avoid Jinja from removing a trailing newline from template strings ::
            DAG(dag_id='my-dag',
                jinja_environment_kwargs={
                    'keep_trailing_newline': True,
                    # some other jinja2 Environment options here
                }
            )
        **See**: `Jinja Environment documentation
        <https://jinja.palletsprojects.com/en/2.11.x/api/#jinja2.Environment>`_
    :param render_template_as_native_obj: If True, uses a Jinja ``NativeEnvironment``
        to render templates as native Python types. If False, a Jinja
        ``Environment`` is used to render templates as string values.
    :param tags: List of tags to help filtering DAGs in the UI.
    :param owner_links: Dict of owners and their links, that will be clickable on the DAGs view UI.
        Can be used as an HTTP link (for example the link to your Slack channel), or a mailto link.
        e.g: {"dag_owner": "https://airflow.apache.org/"}
    :param auto_register: Automatically register this DAG when it is used in a ``with`` block
    :param fail_stop: Fails currently running tasks when task in DAG fails.
        **Warning**: A fail stop dag can only have tasks with the default trigger rule ("all_success").
        An exception will be thrown if any task in a fail stop dag has a non default trigger rule.
    """
    _comps = {
        "dag_id",
        "task_ids",
        "parent_dag",
        "start_date",
        "end_date",
        "schedule_interval",
        "fileloc",