### 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)

fixup_decorator_warning_stack(factory)
        return factory
    return wrapper
STATICA_HACK = True
globals()["kcah_acitats"[::-1].upper()] = False
if STATICA_HACK:  
    from airflow.models.serialized_dag import SerializedDagModel
    DagModel.serialized_dag = relationship(SerializedDagModel)
    """:sphinx-autoapi-skip:"""
class DagContext:
    """
    DAG context is used to keep the current DAG when DAG is used as ContextManager.
    You can use DAG as context:
    .. code-block:: python
        with DAG(
            dag_id="example_dag",
            default_args=default_args,
            schedule="0 0 * * *",
            dagrun_timeout=timedelta(minutes=60),
        ) as dag:
            ...
    If you do this the context stores the DAG and whenever new task is created, it will use
    such stored DAG as the parent DAG.
    """
    _context_managed_dags: collections.deque[DAG] = deque()
    autoregistered_dags: set[tuple[DAG, ModuleType]] = set()
    current_autoregister_module_name: str | None = None
    @classmethod
    def push_context_managed_dag(cls, dag: DAG):