### 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)

run_id=str(i),
                    processor_subdir=str(tmpdir),
                )
                session.add(DbCallbackRequest(callback=callback, priority_weight=i))
        child_pipe, parent_pipe = multiprocessing.Pipe()
        manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory=str(tmpdir),
                max_runs=1,
                processor_timeout=timedelta(days=365),
                signal_conn=child_pipe,
                dag_ids=[],
                pickle_dags=False,
                async_mode=False,
            ),
        )
        with create_session() as session:
            self.run_processor_manager_one_loop(manager, parent_pipe)
            assert session.query(DbCallbackRequest).count() == 3
        with create_session() as session:
            self.run_processor_manager_one_loop(manager, parent_pipe)
            assert session.query(DbCallbackRequest).count() == 1
    @conf_vars(
        {
            ("scheduler", "standalone_dag_processor"): "False",
            ("core", "load_examples"): "False",
        }
    )
    def test_fetch_callbacks_from_database_not_standalone(self, tmpdir):