### 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)

DEFAULT_ARGS = {
        "owner": "test",
        "depends_on_past": True,
        "start_date": timezone.utcnow(),
        "retries": 1,
        "retry_delay": timedelta(minutes=1),
    }
    DEFAULT_DATE = timezone.datetime(2016, 1, 1)
    VALUE = 42
    def setup_method(self):
        self.operator = None
    def teardown_method(self):
        clear_db_runs()
    def test_fileloc(self):
        @dag_decorator(default_args=self.DEFAULT_ARGS)
        def noop_pipeline():
            ...
        dag = noop_pipeline()
        assert isinstance(dag, DAG)
        assert dag.dag_id == "noop_pipeline"
        assert dag.fileloc == __file__
    def test_set_dag_id(self):
        """Test that checks you can set dag_id from decorator."""
        @dag_decorator("test", default_args=self.DEFAULT_ARGS)
        def noop_pipeline():