### 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)

elif not task.start_date:
            task.start_date = self.start_date
        
        
        elif self.start_date:
            task.start_date = max(task.start_date, self.start_date)
        
        if not task.end_date:
            task.end_date = self.end_date
        
        
        elif task.end_date and self.end_date:
            task.end_date = min(task.end_date, self.end_date)
        task_id = task.task_id
        if not task.task_group:
            task_group = TaskGroupContext.get_current_task_group(self)
            if task_group:
                task_id = task_group.child_id(task_id)
                task_group.add(task)
        if (
            task_id in self.task_dict and self.task_dict[task_id] is not task
        ) or task_id in self._task_group.used_group_ids:
            raise DuplicateTaskIdFound(f"Task id '{task_id}' has already been added to the DAG")
        else:
            self.task_dict[task_id] = task
            task.dag = self
            
            self._task_group.used_group_ids.add(task_id)
        self.task_count = len(self.task_dict)
    def add_tasks(self, tasks: Iterable[Operator]) -> None: