### 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)

self.schedule_interval = self.timetable.summary
        else:
            if isinstance(schedule_interval, ArgNotSet):
                schedule_interval = DEFAULT_SCHEDULE_INTERVAL
            self.schedule_interval = schedule_interval
            self.timetable = create_timetable(schedule_interval, self.timezone)
        if isinstance(template_searchpath, str):
            template_searchpath = [template_searchpath]
        self.template_searchpath = template_searchpath
        self.template_undefined = template_undefined
        self.last_loaded = timezone.utcnow()
        self.safe_dag_id = dag_id.replace(".", "__dot__")
        self.max_active_runs = max_active_runs
        if self.timetable.active_runs_limit is not None:
            if self.timetable.active_runs_limit < self.max_active_runs:
                raise AirflowException(
                    f"Invalid max_active_runs: {type(self.timetable)} "
                    f"requires max_active_runs <= {self.timetable.active_runs_limit}"
                )
        self.dagrun_timeout = dagrun_timeout
        self.sla_miss_callback = sla_miss_callback
        if default_view in DEFAULT_VIEW_PRESETS:
            self._default_view: str = default_view
        elif default_view == "tree":
            warnings.warn(
                "`default_view` of 'tree' has been renamed to 'grid' -- please update your DAG",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
            self._default_view = "grid"