### 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)

for ti in test_dag.get_task_instances_before(base_date=REF_DATE, num=0, session=session)
            ]
        )
        assert set([dr.run_id for dr in [dr8]]) == set(
            [
                ti.run_id
                for ti in test_dag.get_task_instances_before(base_date=REF_DATE, num=1, session=session)
            ]
        )
        assert set([dr.run_id for dr in [dr8]]) == set(
            [
                ti.run_id
                for ti in test_dag.get_task_instances_before(base_date=REF_DATE, num=10, session=session)
            ]
        )
        session.close()
    def test_user_defined_filters_macros(self):
        def jinja_udf(name):
            return f"Hello {name}"
        dag = models.DAG(
            "test-dag",
            start_date=DEFAULT_DATE,
            user_defined_filters={"hello": jinja_udf},
            user_defined_macros={"foo": "bar"},
        )
        jinja_env = dag.get_template_env()
        assert "hello" in jinja_env.filters
        assert jinja_env.filters["hello"] == jinja_udf
        assert jinja_env.globals["foo"] == "bar"
    def test_set_jinja_env_additional_option(self):