### 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)

TaskOutletDatasetReference,
        )
        dag_references = collections.defaultdict(set)
        outlet_references = collections.defaultdict(set)
        
        outlet_datasets: dict[Dataset, None] = {}
        input_datasets: dict[Dataset, None] = {}
        
        
        
        
        for dag in dags:
            curr_orm_dag = existing_dags.get(dag.dag_id)
            if not dag.dataset_triggers:
                if curr_orm_dag and curr_orm_dag.schedule_dataset_references:
                    curr_orm_dag.schedule_dataset_references = []
            for dataset in dag.dataset_triggers:
                dag_references[dag.dag_id].add(dataset.uri)
                input_datasets[DatasetModel.from_public(dataset)] = None
            curr_outlet_references = curr_orm_dag and curr_orm_dag.task_outlet_dataset_references
            for task in dag.tasks:
                dataset_outlets = [x for x in task.outlets or [] if isinstance(x, Dataset)]
                if not dataset_outlets:
                    if curr_outlet_references:
                        this_task_outlet_refs = [
                            x
                            for x in curr_outlet_references
                            if x.dag_id == dag.dag_id and x.task_id == task.task_id
                        ]
                        for ref in this_task_outlet_refs: