### 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)

if not hasattr(self, "__data_cache") or self.__data_cache is None:
            if self._data_compressed:
                self.__data_cache = json.loads(zlib.decompress(self._data_compressed))
            else:
                self.__data_cache = self._data
        return self.__data_cache
    @property
    def dag(self) -> SerializedDAG:
        """The DAG deserialized from the ``data`` column."""
        SerializedDAG._load_operator_extra_links = self.load_op_links
        if isinstance(self.data, dict):
            data = self.data
        elif isinstance(self.data, str):
            data = json.loads(self.data)
        else:
            raise ValueError("invalid or missing serialized DAG data")
        return SerializedDAG.from_dict(data)
    @classmethod
    @provide_session
    def remove_dag(cls, dag_id: str, session: Session = NEW_SESSION) -> None:
        """
        Delete a DAG with given dag_id.
        :param dag_id: dag_id to be deleted
        :param session: ORM Session.
        """
        session.execute(cls.__table__.delete().where(cls.dag_id == dag_id))
    @classmethod
    @provide_session
    def remove_deleted_dags(