### 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 contextlib import redirect_stdout
from datetime import timedelta
from pathlib import Path
from tempfile import NamedTemporaryFile
from unittest import mock
from unittest.mock import patch
import jinja2
import pendulum
import pytest
import time_machine
from dateutil.relativedelta import relativedelta
from sqlalchemy import inspect
import airflow
from airflow import models, settings
from airflow.configuration import conf
from airflow.datasets import Dataset
from airflow.decorators import setup, task as task_decorator, teardown
from airflow.exceptions import (
    AirflowException,
    DuplicateTaskIdFound,
    ParamValidationError,
    RemovedInAirflow3Warning,
)
from airflow.models import DAG, DagModel, DagRun, DagTag, TaskFail, TaskInstance as TI
from airflow.models.baseoperator import BaseOperator
from airflow.models.dag import DagOwnerAttributes, dag as dag_decorator, get_dataset_triggered_next_run_info
from airflow.models.dataset import DatasetDagRunQueue, DatasetEvent, DatasetModel, TaskOutletDatasetReference
from airflow.models.param import DagParam, Param, ParamsDict
from airflow.models.serialized_dag import SerializedDagModel
from airflow.operators.bash import BashOperator