### 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)

existing_orm_filelocs = {orm.fileloc for orm in existing_orm_dag_codes_by_fileloc_hashes.values()}
        if not existing_orm_filelocs.issubset(filelocs):
            conflicting_filelocs = existing_orm_filelocs.difference(filelocs)
            hashes_to_filelocs = {DagCode.dag_fileloc_hash(fileloc): fileloc for fileloc in filelocs}
            message = ""
            for fileloc in conflicting_filelocs:
                filename = hashes_to_filelocs[DagCode.dag_fileloc_hash(fileloc)]
                message += (
                    f"Filename '{filename}' causes a hash collision in the "
                    f"database with '{fileloc}'. Please rename the file."
                )
            raise AirflowException(message)
        existing_filelocs = {dag_code.fileloc for dag_code in existing_orm_dag_codes}
        missing_filelocs = filelocs.difference(existing_filelocs)
        for fileloc in missing_filelocs:
            orm_dag_code = DagCode(fileloc, cls._get_code_from_file(fileloc))
            session.add(orm_dag_code)
        for fileloc in existing_filelocs:
            current_version = existing_orm_dag_codes_by_fileloc_hashes[filelocs_to_hashes[fileloc]]
            file_mod_time = datetime.fromtimestamp(
                os.path.getmtime(correct_maybe_zipped(fileloc)), tz=timezone.utc
            )
            if file_mod_time > current_version.last_updated:
                orm_dag_code = existing_orm_dag_codes_map[fileloc]
                orm_dag_code.last_updated = file_mod_time
                orm_dag_code.source_code = cls._get_code_from_file(orm_dag_code.fileloc)
                session.merge(orm_dag_code)
    @classmethod
    @provide_session
    def remove_deleted_code(cls, alive_dag_filelocs: Collection[str], session: Session = NEW_SESSION) -> None: