### 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)

schedule="*/10 * * * *",
            start_date=six_hours_ago_to_the_hour,
            catchup=False,
        )
        next_date, _ = dag1.next_dagrun_info(None)
        
        assert next_date > half_an_hour_ago
        assert next_date < timezone.utcnow()
        dag2 = make_dag(
            dag_id="dag_without_catchup_hourly",
            schedule="@hourly",
            start_date=six_hours_ago_to_the_hour,
            catchup=False,
        )
        next_date, _ = dag2.next_dagrun_info(None)
        
        assert next_date > two_hours_ago
        
        assert next_date < timezone.utcnow()
        dag3 = make_dag(
            dag_id="dag_without_catchup_once",
            schedule="@once",
            start_date=six_hours_ago_to_the_hour,
            catchup=False,
        )
        next_date, _ = dag3.next_dagrun_info(None)
        
        assert next_date == six_hours_ago_to_the_hour
    @time_machine.travel(timezone.datetime(2020, 1, 5), tick=False)
    def test_next_dagrun_info_timedelta_schedule_and_catchup_false(self):