### 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)

"""Verify correctness of dag.tree_view()."""
        with DAG("test_dag", start_date=DEFAULT_DATE) as dag:
            op1 = EmptyOperator(task_id="t1")
            op2 = EmptyOperator(task_id="t2")
            op3 = EmptyOperator(task_id="t3")
            op1 >> op2 >> op3
            with redirect_stdout(io.StringIO()) as stdout:
                dag.tree_view()
                stdout = stdout.getvalue()
            stdout_lines = stdout.split("\n")
            assert "t1" in stdout_lines[0]
            assert "t2" in stdout_lines[1]
            assert "t3" in stdout_lines[2]
    def test_duplicate_task_ids_not_allowed_with_dag_context_manager(self):
        """Verify tasks with Duplicate task_id raises error"""
        with pytest.raises(DuplicateTaskIdFound, match="Task id 't1' has already been added to the DAG"):
            with DAG("test_dag", start_date=DEFAULT_DATE) as dag:
                op1 = EmptyOperator(task_id="t1")
                op2 = BashOperator(task_id="t1", bash_command="sleep 1")
                op1 >> op2
        assert dag.task_dict == {op1.task_id: op1}
    def test_duplicate_task_ids_not_allowed_without_dag_context_manager(self):