### 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)

#
#
#
"""Serialized DAG table in database."""
from __future__ import annotations
import logging
import zlib
from datetime import datetime, timedelta
from typing import Collection
import sqlalchemy_jsonfield
from sqlalchemy import BigInteger, Column, Index, LargeBinary, String, and_, or_, select
from sqlalchemy.orm import Session, backref, foreign, relationship
from sqlalchemy.sql.expression import func, literal
from airflow.models.base import ID_LEN, Base
from airflow.models.dag import DAG, DagModel
from airflow.models.dagcode import DagCode
from airflow.models.dagrun import DagRun
from airflow.serialization.serialized_objects import DagDependency, SerializedDAG
from airflow.settings import COMPRESS_SERIALIZED_DAGS, MIN_SERIALIZED_DAG_UPDATE_INTERVAL, json
from airflow.utils import timezone
from airflow.utils.hashlib_wrapper import md5
from airflow.utils.session import NEW_SESSION, provide_session
from airflow.utils.sqlalchemy import UtcDateTime
log = logging.getLogger(__name__)
class SerializedDagModel(Base):