### 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)

import airflow.models
from airflow.api_internal.internal_api_call import internal_api_call
from airflow.callbacks.callback_requests import CallbackRequest, SlaCallbackRequest
from airflow.configuration import conf
from airflow.dag_processing.processor import DagFileProcessorProcess
from airflow.models import errors
from airflow.models.dag import DagModel
from airflow.models.dagwarning import DagWarning
from airflow.models.db_callback_request import DbCallbackRequest
from airflow.models.serialized_dag import SerializedDagModel
from airflow.secrets.cache import SecretCache
from airflow.stats import Stats
from airflow.utils import timezone
from airflow.utils.file import list_py_file_paths, might_contain_dag
from airflow.utils.log.logging_mixin import LoggingMixin
from airflow.utils.mixins import MultiprocessingStartMethodMixin
from airflow.utils.net import get_hostname
from airflow.utils.process_utils import (
    kill_child_processes_by_pids,
    reap_process_group,
    set_new_process_group,
)
from airflow.utils.retries import retry_db_transaction
from airflow.utils.session import NEW_SESSION, provide_session
from airflow.utils.sqlalchemy import prohibit_commit, skip_locked, with_row_locks
class DagParsingStat(NamedTuple):
    """Information on processing progress."""
    done: bool
    all_files_processed: bool
class DagFileStat(NamedTuple):