### 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)

with create_session() as session:
            
            dag = dagbag.get_dag("test_example_bash_operator")
            dag.last_parsed_time = timezone.utcnow()
            dag.sync_to_db(processor_subdir=dag_directory)
            
            other_dag = other_dagbag.get_dag("test_start_date_scheduling")
            other_dag.last_parsed_time = timezone.utcnow()
            other_dag.sync_to_db(processor_subdir="other")
            
            stat = DagFileStat(
                num_dags=1,
                import_errors=0,
                last_finish_time=timezone.utcnow() + timedelta(hours=1),
                last_duration=1,
                run_count=1,
            )
            manager.processor._file_paths = [test_dag_path]
            manager.processor._file_stats[test_dag_path] = stat
            active_dag_count = session.query(func.count(DagModel.dag_id)).filter(DagModel.is_active).scalar()
            assert active_dag_count == 2
            manager.processor._scan_stale_dags()
            active_dag_count = session.query(func.count(DagModel.dag_id)).filter(DagModel.is_active).scalar()
            assert active_dag_count == 1
    @mock.patch(
        "airflow.dag_processing.processor.DagFileProcessorProcess.waitable_handle", new_callable=PropertyMock
    )
    @mock.patch("airflow.dag_processing.processor.DagFileProcessorProcess.pid", new_callable=PropertyMock)
    @mock.patch("airflow.dag_processing.processor.DagFileProcessorProcess.kill")
    def test_kill_timed_out_processors_kill(self, mock_kill, mock_pid, mock_waitable_handle):