### 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)

memo.setdefault(id(t.task_group), None)
            return copy.deepcopy(t, memo)
        dag.task_dict = {
            t.task_id: _deepcopy_task(t)
            for t in itertools.chain(matched_tasks, also_include, direct_upstreams)
        }
        def filter_task_group(group, parent_group):
            """Exclude tasks not included in the subdag from the given TaskGroup."""
            
            
            
            copied = copy.copy(group)
            memo[id(group.children)] = {}
            if parent_group:
                memo[id(group.parent_group)] = parent_group
            for attr, value in copied.__dict__.items():
                if id(value) in memo:
                    value = memo[id(value)]
                else:
                    value = copy.deepcopy(value, memo)
                copied.__dict__[attr] = value
            proxy = weakref.proxy(copied)
            for child in group.children.values():