### 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)

return f"<SerializedDag: {self.dag_id}>"
    @classmethod
    @provide_session
    def write_dag(
        cls,
        dag: DAG,
        min_update_interval: int | None = None,
        processor_subdir: str | None = None,
        session: Session = NEW_SESSION,
    ) -> bool:
        """
        Serialize a DAG and writes it into database.
        If the record already exists, it checks if the Serialized DAG changed or not. If it is
        changed, it updates the record, ignores otherwise.
        :param dag: a DAG to be written into database
        :param min_update_interval: minimal interval in seconds to update serialized DAG
        :param session: ORM Session
        :returns: Boolean indicating if the DAG was written to the DB
        """