### 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 list(self.task_dict.values())
    @tasks.setter
    def tasks(self, val):
        raise AttributeError("DAG.tasks can not be modified. Use dag.add_task() instead.")
    @property
    def task_ids(self) -> list[str]:
        return list(self.task_dict)
    @property
    def teardowns(self) -> list[Operator]:
        return [task for task in self.tasks if getattr(task, "is_teardown", None)]
    @property
    def tasks_upstream_of_teardowns(self) -> list[Operator]:
        upstream_tasks = [t.upstream_list for t in self.teardowns]
        return [val for sublist in upstream_tasks for val in sublist if not getattr(val, "is_teardown", None)]
    @property
    def task_group(self) -> TaskGroup:
        return self._task_group
    @property
    def filepath(self) -> str:
        """Relative file path to the DAG.
        :meta private:
        """
        warnings.warn(
            "filepath is deprecated, use relative_fileloc instead",
            RemovedInAirflow3Warning,
            stacklevel=2,
        )
        return str(self.relative_fileloc)
    @property
    def relative_fileloc(self) -> pathlib.Path: