### 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 task_id in self.task_dict
    def has_task_group(self, task_group_id: str) -> bool:
        return task_group_id in self.task_group_dict
    @functools.cached_property
    def task_group_dict(self):
        return {k: v for k, v in self._task_group.get_task_group_dict().items() if k is not None}
    def get_task(self, task_id: str, include_subdags: bool = False) -> Operator:
        if task_id in self.task_dict:
            return self.task_dict[task_id]
        if include_subdags:
            for dag in self.subdags:
                if task_id in dag.task_dict:
                    return dag.task_dict[task_id]
        raise TaskNotFound(f"Task {task_id} not found")
    def pickle_info(self):