### 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)

clear_db_serialized_dags()
        clear_db_dags()
        test_dag_path = TEST_DAG_FOLDER / "test_scheduler_dags.py"
        async_mode = "sqlite" not in conf.get("database", "sql_alchemy_conn")
        processor_agent = DagFileProcessorAgent(test_dag_path, 1, timedelta(days=365), [], False, async_mode)
        processor_agent.start()
        if not async_mode:
            processor_agent.run_single_parsing_loop()
        while not processor_agent.done:
            if not async_mode:
                processor_agent.wait_until_finished()
            processor_agent.heartbeat()
        assert processor_agent.all_files_processed
        assert processor_agent.done
        with create_session() as session:
            dag_ids = session.query(DagModel.dag_id).order_by("dag_id").all()
            assert dag_ids == [("test_start_date_scheduling",), ("test_task_start_date_scheduling",)]
            dag_ids = session.query(SerializedDagModel.dag_id).order_by("dag_id").all()
            assert dag_ids == [("test_start_date_scheduling",), ("test_task_start_date_scheduling",)]
    def test_launch_process(self):