### 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)

matched_tasks = [t for t in self.tasks if t.task_id in task_ids_or_regex]
        also_include_ids: set[str] = set()
        for t in matched_tasks:
            if include_downstream:
                for rel in t.get_flat_relatives(upstream=False):
                    also_include_ids.add(rel.task_id)
                    if rel not in matched_tasks:  
                        
                        
                        if not rel.is_setup and not rel.is_teardown:
                            also_include_ids.update(
                                x.task_id for x in rel.get_upstreams_only_setups_and_teardowns()
                            )
            if include_upstream:
                also_include_ids.update(x.task_id for x in t.get_upstreams_follow_setups())
            else:
                if not t.is_setup and not t.is_teardown:
                    also_include_ids.update(x.task_id for x in t.get_upstreams_only_setups_and_teardowns())
            if t.is_setup and not include_downstream:
                also_include_ids.update(x.task_id for x in t.downstream_list if x.is_teardown)
        also_include: list[Operator] = [self.task_dict[x] for x in also_include_ids]
        direct_upstreams: list[Operator] = []
        if include_direct_upstream:
            for t in itertools.chain(matched_tasks, also_include):
                upstream = (u for u in t.upstream_list if isinstance(u, (BaseOperator, MappedOperator)))
                direct_upstreams.extend(upstream)
        
        
        
        def _deepcopy_task(t) -> Operator: