### 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)

RemovedInAirflow3Warning,
    TaskNotFound,
)
from airflow.jobs.job import run_job
from airflow.models.abstractoperator import AbstractOperator
from airflow.models.base import Base, StringID
from airflow.models.baseoperator import BaseOperator
from airflow.models.dagcode import DagCode
from airflow.models.dagpickle import DagPickle
from airflow.models.dagrun import RUN_ID_REGEX, DagRun
from airflow.models.operator import Operator
from airflow.models.param import DagParam, ParamsDict
from airflow.models.taskinstance import Context, TaskInstance, TaskInstanceKey, clear_task_instances
from airflow.secrets.local_filesystem import LocalFilesystemBackend
from airflow.security import permissions
from airflow.stats import Stats
from airflow.timetables.base import DagRunInfo, DataInterval, TimeRestriction, Timetable
from airflow.timetables.interval import CronDataIntervalTimetable, DeltaDataIntervalTimetable
from airflow.timetables.simple import (
    ContinuousTimetable,
    DatasetTriggeredTimetable,
    NullTimetable,
    OnceTimetable,
)
from airflow.typing_compat import Literal
from airflow.utils import timezone
from airflow.utils.dag_cycle_tester import check_cycle
from airflow.utils.dates import cron_presets, date_range as utils_date_range
from airflow.utils.decorators import fixup_decorator_warning_stack
from airflow.utils.helpers import at_most_one, exactly_one, validate_key