### 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 "full" paths to DAGs if inside ZIP files.
                This is the format used by the remove/delete functions.
                """
                if fileloc.endswith(".py") or not zipfile.is_zipfile(fileloc):
                    yield fileloc
                    return
                try:
                    with zipfile.ZipFile(fileloc) as z:
                        for info in z.infolist():
                            if might_contain_dag(info.filename, True, z):
                                yield os.path.join(fileloc, info.filename)
                except zipfile.BadZipFile:
                    self.log.exception("There was an error accessing ZIP file %s %s", fileloc)
            dag_filelocs = {full_loc for path in self._file_paths for full_loc in _iter_dag_filelocs(path)}
            from airflow.models.dagcode import DagCode
            SerializedDagModel.remove_deleted_dags(
                alive_dag_filelocs=dag_filelocs,
                processor_subdir=self.get_dag_directory(),
            )
            DagModel.deactivate_deleted_dags(dag_filelocs)
            DagCode.remove_deleted_code(dag_filelocs)
            return True
        return False
    def _print_stat(self):