### 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)

def setup_method(self):
        dictConfig(DEFAULT_LOGGING_CONFIG)
        clear_db_runs()
        clear_db_serialized_dags()
        clear_db_dags()
        clear_db_callbacks()
    def teardown_class(self):
        clear_db_runs()
        clear_db_serialized_dags()
        clear_db_dags()
        clear_db_callbacks()
    def run_processor_manager_one_loop(self, manager, parent_pipe):
        if not manager.processor._async_mode:
            parent_pipe.send(DagParsingSignal.AGENT_RUN_ONCE)
        results = []
        while True:
            manager.processor._run_parsing_loop()
            while parent_pipe.poll(timeout=0.01):
                obj = parent_pipe.recv()
                if not isinstance(obj, DagParsingStat):
                    results.append(obj)
                elif obj.done:
                    return results
            raise RuntimeError("Shouldn't get here - nothing to read, but manager not finished!")
    @conf_vars({("core", "load_examples"): "False"})
    def test_remove_file_clears_import_error(self, tmpdir):