### 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)

return num
            assert isinstance(value, DagParam)
            xcom_arg = return_num(value)
            self.operator = xcom_arg.operator
        dag = xcom_pass_to_op()
        new_value = 52
        dr = dag.create_dagrun(
            run_id=DagRunType.MANUAL.value,
            start_date=timezone.utcnow(),
            execution_date=self.DEFAULT_DATE,
            data_interval=(self.DEFAULT_DATE, self.DEFAULT_DATE),
            state=State.RUNNING,
            conf={"value": new_value},
        )
        self.operator.run(start_date=self.DEFAULT_DATE, end_date=self.DEFAULT_DATE)
        ti = dr.get_task_instances()[0]
        assert ti.xcom_pull() == new_value
    @pytest.mark.parametrize("value", [VALUE, 0])
    def test_set_params_for_dag(self, value):
        """Test that dag param is correctly set when using dag decorator"""
        @dag_decorator(default_args=self.DEFAULT_ARGS)
        def xcom_pass_to_op(value=value):
            @task_decorator
            def return_num(num):
                return num
            xcom_arg = return_num(value)
            self.operator = xcom_arg.operator
        dag = xcom_pass_to_op()
        assert dag.params["value"] == value
    def test_warning_location(self):