### 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)

)
        
        last_finish_time = freezed_base_time - timedelta(seconds=10)
        manager.processor._file_stats = {
            "file_1.py": DagFileStat(1, 0, last_finish_time, timedelta(seconds=1.0), 1),
        }
        with time_machine.travel(freezed_base_time):
            manager.processor.set_file_paths(dag_files)
            assert manager.processor._file_path_queue == collections.deque()
            
            manager.processor.prepare_file_path_queue()
            assert manager.processor._file_path_queue == collections.deque()
        
        
        file_1_new_mtime = freezed_base_time - timedelta(seconds=5)
        file_1_new_mtime_ts = file_1_new_mtime.timestamp()
        with time_machine.travel(freezed_base_time):
            manager.processor.set_file_paths(dag_files)
            assert manager.processor._file_path_queue == collections.deque()
            
            mock_getmtime.side_effect = [file_1_new_mtime_ts]
            manager.processor.prepare_file_path_queue()
            
            assert manager.processor._file_path_queue == collections.deque(["file_1.py"])
            assert last_finish_time < file_1_new_mtime
            assert (
                manager.processor._file_process_interval
                > (freezed_base_time - manager.processor.get_last_finish_time("file_1.py")).total_seconds()
            )
    def test_scan_stale_dags(self):