### 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_or_regex: str | Pattern | Iterable[str],
        include_downstream=False,
        include_upstream=True,
        include_direct_upstream=False,
    ):
        """
        Return a subset of the current dag based on regex matching one or more tasks.
        Returns a subset of the current dag as a deep copy of the current dag
        based on a regex that should match one or many tasks, and includes
        upstream and downstream neighbours based on the flag passed.
        :param task_ids_or_regex: Either a list of task_ids, or a regex to
            match against task ids (as a string, or compiled regex pattern).
        :param include_downstream: Include all downstream tasks of matched
            tasks, in addition to matched tasks.
        :param include_upstream: Include all upstream tasks of matched tasks,
            in addition to matched tasks.
        :param include_direct_upstream: Include all tasks directly upstream of matched
            and downstream (if include_downstream = True) tasks
        """
        from airflow.models.baseoperator import BaseOperator
        from airflow.models.mappedoperator import MappedOperator
        
        
        memo = {id(self.task_dict): None, id(self._task_group): None}
        dag = copy.deepcopy(self, memo)  
        if isinstance(task_ids_or_regex, (str, Pattern)):
            matched_tasks = [t for t in self.tasks if re2.findall(task_ids_or_regex, t.task_id)]
        else: