### 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)

"""File location of the importable dag 'file' relative to the configured DAGs folder."""
        path = pathlib.Path(self.fileloc)
        try:
            rel_path = path.relative_to(self._processor_dags_folder or settings.DAGS_FOLDER)
            if rel_path == pathlib.Path("."):
                return path
            else:
                return rel_path
        except ValueError:
            
            return path
    @property
    def folder(self) -> str:
        """Folder location of where the DAG object is instantiated."""
        return os.path.dirname(self.fileloc)
    @property
    def owner(self) -> str:
        """
        Return list of all owners found in DAG tasks.
        :return: Comma separated list of owners in DAG tasks
        """
        return ", ".join({t.owner for t in self.tasks})
    @property
    def allow_future_exec_dates(self) -> bool: