### 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)

ti2.state = State.RUNNING
        ti3 = TI(task=test_task, run_id=dr3.run_id)
        ti3.state = State.QUEUED
        ti4 = TI(task=test_task, run_id=dr4.run_id)
        ti4.state = State.RUNNING
        session = settings.Session()
        session.merge(ti1)
        session.merge(ti2)
        session.merge(ti3)
        session.merge(ti4)
        session.commit()
        assert 0 == DAG.get_num_task_instances(test_dag_id, task_ids=["fakename"], session=session)
        assert 4 == DAG.get_num_task_instances(test_dag_id, task_ids=[test_task_id], session=session)
        assert 4 == DAG.get_num_task_instances(
            test_dag_id, task_ids=["fakename", test_task_id], session=session
        )
        assert 1 == DAG.get_num_task_instances(
            test_dag_id, task_ids=[test_task_id], states=[None], session=session
        )
        assert 2 == DAG.get_num_task_instances(
            test_dag_id, task_ids=[test_task_id], states=[State.RUNNING], session=session
        )
        assert 3 == DAG.get_num_task_instances(
            test_dag_id, task_ids=[test_task_id], states=[None, State.RUNNING], session=session
        )
        assert 4 == DAG.get_num_task_instances(
            test_dag_id, task_ids=[test_task_id], states=[None, State.QUEUED, State.RUNNING], session=session
        )
        session.close()
    def test_get_task_instances_before(self):