### 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)

with assert_queries_count(9):
            DAG.bulk_write_to_db(dags)
        with create_session() as session:
            assert {"dag-bulk-sync-0", "dag-bulk-sync-1", "dag-bulk-sync-2", "dag-bulk-sync-3"} == {
                row[0] for row in session.query(DagModel.dag_id).all()
            }
            assert {
                ("dag-bulk-sync-0", "test-dag"),
                ("dag-bulk-sync-0", "test-dag2"),
                ("dag-bulk-sync-1", "test-dag"),
                ("dag-bulk-sync-1", "test-dag2"),
                ("dag-bulk-sync-2", "test-dag"),
                ("dag-bulk-sync-2", "test-dag2"),
                ("dag-bulk-sync-3", "test-dag"),
                ("dag-bulk-sync-3", "test-dag2"),
            } == set(session.query(DagTag.dag_id, DagTag.name).all())
        
        for dag in dags:
            dag.tags.remove("test-dag")
        with assert_queries_count(9):
            DAG.bulk_write_to_db(dags)
        with create_session() as session:
            assert {"dag-bulk-sync-0", "dag-bulk-sync-1", "dag-bulk-sync-2", "dag-bulk-sync-3"} == {
                row[0] for row in session.query(DagModel.dag_id).all()
            }
            assert {
                ("dag-bulk-sync-0", "test-dag2"),
                ("dag-bulk-sync-1", "test-dag2"),
                ("dag-bulk-sync-2", "test-dag2"),
                ("dag-bulk-sync-3", "test-dag2"),