### 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)

DagParsingSignal,
    DagParsingStat,
)
from airflow.dag_processing.processor import DagFileProcessorProcess
from airflow.jobs.dag_processor_job_runner import DagProcessorJobRunner
from airflow.jobs.job import Job
from airflow.models import DagBag, DagModel, DbCallbackRequest, errors
from airflow.models.dagcode import DagCode
from airflow.models.serialized_dag import SerializedDagModel
from airflow.utils import timezone
from airflow.utils.net import get_hostname
from airflow.utils.session import create_session
from tests.core.test_logging_config import SETTINGS_FILE_VALID, settings_context
from tests.models import TEST_DAGS_FOLDER
from tests.test_utils.config import conf_vars
from tests.test_utils.db import clear_db_callbacks, clear_db_dags, clear_db_runs, clear_db_serialized_dags
TEST_DAG_FOLDER = pathlib.Path(__file__).parents[1].resolve() / "dags"
DEFAULT_DATE = timezone.datetime(2016, 1, 1)
class FakeDagFileProcessorRunner(DagFileProcessorProcess):
    
    
    def __init__(self, file_path, pickle_dags, dag_ids, dag_directory, callbacks):
        super().__init__(file_path, pickle_dags, dag_ids, dag_directory, callbacks)
        
        readable, writable = multiprocessing.Pipe(duplex=False)
        writable.send("abc")
        writable.close()
        self._waitable_handle = readable
        self._result = 0, 0
    def start(self):