### 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)

dag = DAG("test-dag", jinja_environment_kwargs={"keep_trailing_newline": True, "cache_size": 50})
        jinja_env = dag.get_template_env()
        assert jinja_env.keep_trailing_newline is True
        assert jinja_env.cache.capacity == 50
        assert jinja_env.undefined is jinja2.StrictUndefined
    def test_template_undefined(self):
        dag = DAG("test-dag", template_undefined=jinja2.Undefined)
        jinja_env = dag.get_template_env()
        assert jinja_env.undefined is jinja2.Undefined
    @pytest.mark.parametrize(
        "use_native_obj, force_sandboxed, expected_env",
        [
            (False, True, SandboxedEnvironment),
            (False, False, SandboxedEnvironment),
            (True, False, NativeEnvironment),
            (True, True, SandboxedEnvironment),
        ],
    )
    def test_template_env(self, use_native_obj, force_sandboxed, expected_env):