### 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)

:param run_id: ID of the DAG run to get the task concurrency of
        :param task_ids: A list of valid task IDs for the given DAG
        :param states: A list of states to filter by if supplied
        :return: The number of running tasks
        """
        qry = select(func.count(TaskInstance.task_id)).where(
            TaskInstance.dag_id == dag_id,
        )
        if run_id:
            qry = qry.where(
                TaskInstance.run_id == run_id,
            )
        if task_ids:
            qry = qry.where(
                TaskInstance.task_id.in_(task_ids),
            )
        if states:
            if None in states:
                if all(x is None for x in states):
                    qry = qry.where(TaskInstance.state.is_(None))
                else:
                    not_none_states = [state for state in states if state]
                    qry = qry.where(
                        or_(TaskInstance.state.in_(not_none_states), TaskInstance.state.is_(None))
                    )
            else:
                qry = qry.where(TaskInstance.state.in_(states))
        return session.scalar(qry)
    @classmethod
    def get_serialized_fields(cls):