### 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)

"""Read all DAGs in serialized_dag table.
        :param session: ORM Session
        :returns: a dict of DAGs read from database
        """
        serialized_dags = session.scalars(select(cls))
        dags = {}
        for row in serialized_dags:
            log.debug("Deserializing DAG: %s", row.dag_id)
            dag = row.dag
            
            if dag.dag_id == row.dag_id:
                dags[row.dag_id] = dag
            else:
                log.warning(
                    "dag_id Mismatch in DB: Row with dag_id '%s' has Serialised DAG with '%s' dag_id",
                    row.dag_id,
                    dag.dag_id,
                )
        return dags
    @property
    def data(self) -> dict | None: