### 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)

if start_date and start_date.tzinfo:
            tzinfo = None if start_date.tzinfo else settings.TIMEZONE
            tz = pendulum.instance(start_date, tz=tzinfo).timezone
        elif "start_date" in self.default_args and self.default_args["start_date"]:
            date = self.default_args["start_date"]
            if not isinstance(date, datetime):
                date = timezone.parse(date)
                self.default_args["start_date"] = date
                start_date = date
            tzinfo = None if date.tzinfo else settings.TIMEZONE
            tz = pendulum.instance(date, tz=tzinfo).timezone
        self.timezone: Timezone = tz or settings.TIMEZONE
        
        if "end_date" in self.default_args and self.default_args["end_date"]:
            if isinstance(self.default_args["end_date"], str):
                self.default_args["end_date"] = timezone.parse(
                    self.default_args["end_date"], timezone=self.timezone
                )
        self.start_date = timezone.convert_to_utc(start_date)
        self.end_date = timezone.convert_to_utc(end_date)
        
        if "start_date" in self.default_args:
            self.default_args["start_date"] = timezone.convert_to_utc(self.default_args["start_date"])
        if "end_date" in self.default_args:
            self.default_args["end_date"] = timezone.convert_to_utc(self.default_args["end_date"])
        
        scheduling_args = [schedule_interval, timetable, schedule]
        if not at_most_one(*scheduling_args):
            raise ValueError("At most one allowed for args 'schedule_interval', 'timetable', and 'schedule'.")
        if schedule_interval is not NOTSET: