### 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)

dag = DAG("dag_paused")
        dag.sync_to_db()
        assert not dag.get_is_paused()
        dag = DAG("dag_paused", is_paused_upon_creation=True)
        dag.sync_to_db()
        
        assert not dag.get_is_paused()
    def test_new_dag_is_paused_upon_creation(self):
        dag = DAG("new_nonexisting_dag", is_paused_upon_creation=True)
        session = settings.Session()
        dag.sync_to_db(session=session)
        orm_dag = session.query(DagModel).filter(DagModel.dag_id == "new_nonexisting_dag").one()
        
        assert orm_dag.is_paused
        session.close()
    def test_existing_dag_default_view(self):