### 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)

self,
        *,
        task_ids: Collection[str | tuple[str, int]] | None,
        as_pk_tuple: Literal[True, None] = None,
        start_date: datetime | None,
        end_date: datetime | None,
        run_id: str | None,
        state: TaskInstanceState | Sequence[TaskInstanceState],
        include_subdags: bool,
        include_parentdag: bool,
        include_dependent_dags: bool,
        exclude_task_ids: Collection[str | tuple[str, int]] | None,
        session: Session,
        dag_bag: DagBag | None = None,
        recursion_depth: int = 0,
        max_recursion_depth: int | None = None,
        visited_external_tis: set[TaskInstanceKey] | None = None,
    ) -> Iterable[TaskInstance] | set[TaskInstanceKey]:
        TI = TaskInstance