### 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)

"""
        Test that when 'params' exists as a key passed to the default_args dict
        in addition to params being passed explicitly as an argument to the
        dag, that the 'params' key of the default_args dict is merged with the
        dict of the params argument.
        """
        params1 = {"parameter1": 1}
        params2 = {"parameter2": 2}
        dag = models.DAG("test-dag", default_args={"params": params1}, params=params2)
        assert params1["parameter1"] == dag.params["parameter1"]
        assert params2["parameter2"] == dag.params["parameter2"]
    def test_not_none_schedule_with_non_default_params(self):
        """
        Test if there is a DAG with not None schedule_interval and have some params that
        don't have a default value raise a error while DAG parsing
        """
        params = {"param1": Param(type="string")}
        with pytest.raises(AirflowException):
            models.DAG("dummy-dag", params=params)
    def test_dag_invalid_default_view(self):
        """
        Test invalid `default_view` of DAG initialization
        """
        with pytest.raises(AirflowException, match="Invalid values of dag.default_view: only support"):
            models.DAG(dag_id="test-invalid-default_view", default_view="airflow")
    def test_dag_default_view_default_value(self):