### 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)

"""Test that @dag uses function docs as doc_md for DAG object"""
        @dag_decorator(default_args=self.DEFAULT_ARGS)
        def noop_pipeline():
            """
            {% if True %}
               Regular DAG documentation
            {% endif %}
            """
        dag = noop_pipeline()
        assert isinstance(dag, DAG)
        assert dag.dag_id == "noop_pipeline"
        assert "Regular DAG documentation" in dag.doc_md
    def test_resolve_documentation_template_file_rendered(self):
        """Test that @dag uses function docs as doc_md for DAG object"""
        with NamedTemporaryFile(suffix=".md") as f:
            f.write(
                b"""
            {% if True %}
               External Markdown DAG documentation
            {% endif %}
            """
            )
            f.flush()
            template_dir = os.path.dirname(f.name)
            template_file = os.path.basename(f.name)
            @dag_decorator(
                "test-dag", start_date=DEFAULT_DATE, template_searchpath=template_dir, doc_md=template_file
            )
            def markdown_docs():