### 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)

assert not SerializedDagModel.has_dag("test_zip_dag")
        
        assert not DagCode.has_dag(dag.fileloc)
        
        assert not dag.get_is_active()
    @conf_vars(
        {
            ("core", "load_examples"): "False",
            ("scheduler", "standalone_dag_processor"): "True",
        }
    )
    def test_fetch_callbacks_from_database(self, tmpdir):
        """Test DagProcessorJobRunner._fetch_callbacks method"""
        dag_filepath = TEST_DAG_FOLDER / "test_on_failure_callback_dag.py"
        callback1 = DagCallbackRequest(
            dag_id="test_start_date_scheduling",
            full_filepath=str(dag_filepath),
            is_failure_callback=True,
            processor_subdir=str(tmpdir),
            run_id="123",
        )
        callback2 = DagCallbackRequest(
            dag_id="test_start_date_scheduling",
            full_filepath=str(dag_filepath),
            is_failure_callback=True,
            processor_subdir=str(tmpdir),
            run_id="456",
        )
        callback3 = SlaCallbackRequest(
            dag_id="test_start_date_scheduling",