### 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)

"""
        Add a list of tasks to the DAG.
        :param tasks: a lit of tasks you want to add
        """
        for task in tasks:
            self.add_task(task)
    def _remove_task(self, task_id: str) -> None:
        
        
        task = self.task_dict.pop(task_id)
        tg = getattr(task, "task_group", None)
        if tg:
            tg._remove(task)
        self.task_count = len(self.task_dict)
    def run(
        self,
        start_date=None,
        end_date=None,
        mark_success=False,
        local=False,
        executor=None,
        donot_pickle=airflow_conf.getboolean("core", "donot_pickle"),
        ignore_task_deps=False,
        ignore_first_depends_on_past=True,
        pool=None,