### 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_arg, expected_timetable, interval_description
    ):
        dag = DAG("test_schedule_interval_arg", schedule=schedule_interval_arg)
        assert dag.timetable == expected_timetable
        assert dag.schedule_interval == schedule_interval_arg
        assert dag.timetable.description == interval_description
    def test_timetable_and_description_from_dataset(self):
        dag = DAG("test_schedule_interval_arg", schedule=[Dataset(uri="hello")])
        assert dag.timetable == DatasetTriggeredTimetable()
        assert dag.schedule_interval == "Dataset"
        assert dag.timetable.description == "Triggered by datasets"
    def test_schedule_interval_still_works(self):