### 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.utils.log.logging_mixin import LoggingMixin
from airflow.utils.session import NEW_SESSION, provide_session
from airflow.utils.sqlalchemy import (
    Interval,
    UtcDateTime,
    lock_rows,
    skip_locked,
    tuple_in_condition,
    with_row_locks,
)
from airflow.utils.state import DagRunState, TaskInstanceState
from airflow.utils.types import NOTSET, ArgNotSet, DagRunType, EdgeInfoType
if TYPE_CHECKING:
    from types import ModuleType
    from airflow.datasets import Dataset
    from airflow.decorators import TaskDecoratorCollection
    from airflow.models.dagbag import DagBag
    from airflow.models.slamiss import SlaMiss
    from airflow.utils.task_group import TaskGroup
log = logging.getLogger(__name__)
DEFAULT_VIEW_PRESETS = ["grid", "graph", "duration", "gantt", "landing_times"]
ORIENTATION_PRESETS = ["LR", "TB", "RL", "BT"]
TAG_MAX_LEN = 100
DagStateChangeCallback = Callable[[Context], None]
ScheduleInterval = Union[None, str, timedelta, relativedelta]
ScheduleIntervalArg = Union[ArgNotSet, ScheduleInterval]
ScheduleArg = Union[ArgNotSet, ScheduleInterval, Timetable, Collection["Dataset"]]
SLAMissCallback = Callable[["DAG", str, str, List["SlaMiss"], List[TaskInstance]], None]
DEFAULT_SCHEDULE_INTERVAL = timedelta(days=1)
class InconsistentDataInterval(AirflowException):