### 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)

"""Get the Default DAG View, returns the default config value if DagModel does not have a value."""
        
        return self.default_view or airflow_conf.get_mandatory_value("webserver", "dag_default_view").lower()
    @property
    def safe_dag_id(self):
        return self.dag_id.replace(".", "__dot__")
    @property
    def relative_fileloc(self) -> pathlib.Path | None:
        """File location of the importable dag 'file' relative to the configured DAGs folder."""
        if self.fileloc is None:
            return None
        path = pathlib.Path(self.fileloc)
        try:
            return path.relative_to(settings.DAGS_FOLDER)
        except ValueError:
            
            return path
    @provide_session
    def set_is_paused(self, is_paused: bool, including_subdags: bool = True, session=NEW_SESSION) -> None: