### 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)

...
        dag = noop_pipeline()
        assert isinstance(dag, DAG)
        assert dag.dag_id == "test"
    def test_default_dag_id(self):
        """Test that @dag uses function name as default dag id."""
        @dag_decorator(default_args=self.DEFAULT_ARGS)
        def noop_pipeline():
            ...
        dag = noop_pipeline()
        assert isinstance(dag, DAG)
        assert dag.dag_id == "noop_pipeline"
    @pytest.mark.parametrize(
        argnames=["dag_doc_md", "expected_doc_md"],
        argvalues=[
            pytest.param("dag docs.", "dag docs.", id="use_dag_doc_md"),
            pytest.param(None, "Regular DAG documentation", id="use_dag_docstring"),
        ],
    )
    def test_documentation_added(self, dag_doc_md, expected_doc_md):
        """Test that @dag uses function docs as doc_md for DAG object if doc_md is not explicitly set."""
        @dag_decorator(default_args=self.DEFAULT_ARGS, doc_md=dag_doc_md)
        def noop_pipeline():
            """Regular DAG documentation"""
        dag = noop_pipeline()
        assert isinstance(dag, DAG)
        assert dag.dag_id == "noop_pipeline"
        assert dag.doc_md == expected_doc_md
    def test_documentation_template_rendered(self):