### 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)

assert calculated_weight == correct_weight
    def test_dag_task_priority_weight_total_using_absolute(self):
        
        weight = 10
        width = 5
        depth = 5
        with DAG("dag", start_date=DEFAULT_DATE, default_args={"owner": "owner1"}) as dag:
            pipeline = [
                [
                    EmptyOperator(
                        task_id=f"stage{i}.{j}",
                        priority_weight=weight,
                        weight_rule=WeightRule.ABSOLUTE,
                    )
                    for j in range(0, width)
                ]
                for i in range(0, depth)
            ]
            for i, stage in enumerate(pipeline):
                if i == 0:
                    continue
                for current_task in stage:
                    for prev_task in pipeline[i - 1]:
                        current_task.set_upstream(prev_task)
            for task in dag.task_dict.values():
                
                correct_weight = weight
                calculated_weight = task.priority_weight_total
                assert calculated_weight == correct_weight
    def test_dag_task_invalid_weight_rule(self):