### 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)

[dag2_req1.full_filepath, dag1_req1.full_filepath]
        )
        assert set(manager.processor._callback_to_execute.keys()) == {
            dag1_req1.full_filepath,
            dag2_req1.full_filepath,
        }
        assert manager.processor._callback_to_execute[dag1_req1.full_filepath] == [dag1_req1, dag1_sla1]
        assert manager.processor._callback_to_execute[dag2_req1.full_filepath] == [dag2_req1]
        
        manager.processor._add_callback_to_queue(dag1_sla2)
        manager.processor._add_callback_to_queue(dag3_sla1)
        
        
        assert manager.processor._file_path_queue == collections.deque(
            [dag2_req1.full_filepath, dag1_req1.full_filepath]
        )
        assert manager.processor._callback_to_execute[dag1_req1.full_filepath] == [dag1_req1, dag1_sla1]
        assert manager.processor._callback_to_execute[dag3_sla1.full_filepath] == [dag3_sla1]
        
        manager.processor._add_callback_to_queue(dag1_req2)
        
        assert manager.processor._file_path_queue == collections.deque(
            [dag1_req1.full_filepath, dag2_req1.full_filepath]
        )
        assert manager.processor._callback_to_execute[dag1_req1.full_filepath] == [
            dag1_req1,
            dag1_sla1,
            dag1_req2,
        ]
class TestDagFileProcessorAgent: