### 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)

self.doc_md = self.get_doc_md(doc_md)
        self.tags = tags or []
        self._task_group = TaskGroup.create_root(self)
        self.validate_schedule_and_params()
        wrong_links = dict(self.iter_invalid_owner_links())
        if wrong_links:
            raise AirflowException(
                "Wrong link format was used for the owner. Use a valid link \n"
                f"Bad formatted links are: {wrong_links}"
            )
        
        
        
        self._processor_dags_folder = None
    def get_doc_md(self, doc_md: str | None) -> str | None:
        if doc_md is None:
            return doc_md
        env = self.get_template_env(force_sandboxed=True)
        if not doc_md.endswith(".md"):
            template = jinja2.Template(doc_md)
        else:
            try:
                template = env.get_template(doc_md)
            except jinja2.exceptions.TemplateNotFound:
                return f"""
                # Templating Error!
                Not able to find the template file: `{doc_md}`.
                """
        return template.render()
    def _check_schedule_interval_matches_timetable(self) -> bool: