### 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)

from airflow.operators.empty import EmptyOperator
from airflow.operators.python import PythonOperator
from airflow.operators.subdag import SubDagOperator
from airflow.security import permissions
from airflow.templates import NativeEnvironment, SandboxedEnvironment
from airflow.timetables.base import DagRunInfo, DataInterval, TimeRestriction, Timetable
from airflow.timetables.simple import (
    ContinuousTimetable,
    DatasetTriggeredTimetable,
    NullTimetable,
    OnceTimetable,
)
from airflow.utils import timezone
from airflow.utils.file import list_py_file_paths
from airflow.utils.session import create_session, provide_session
from airflow.utils.state import DagRunState, State, TaskInstanceState
from airflow.utils.task_group import TaskGroup, TaskGroupContext
from airflow.utils.timezone import datetime as datetime_tz
from airflow.utils.types import DagRunType
from airflow.utils.weight_rule import WeightRule
from tests.models import DEFAULT_DATE
from tests.test_utils.asserts import assert_queries_count
from tests.test_utils.config import conf_vars
from tests.test_utils.db import clear_db_dags, clear_db_datasets, clear_db_runs, clear_db_serialized_dags
from tests.test_utils.mapping import expand_mapped_task
from tests.test_utils.timetables import cron_timetable, delta_timetable
TEST_DATE = datetime_tz(2015, 1, 2, 0, 0)
repo_root = Path(airflow.__file__).parent.parent
@pytest.fixture
def clear_dags():