### 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)

self._max_active_tasks = value
    @property
    def access_control(self):
        return self._access_control
    @access_control.setter
    def access_control(self, value):
        self._access_control = DAG._upgrade_outdated_dag_access_control(value)
    @property
    def description(self) -> str | None:
        return self._description
    @property
    def default_view(self) -> str:
        return self._default_view
    @property
    def pickle_id(self) -> int | None:
        return self._pickle_id
    @pickle_id.setter
    def pickle_id(self, value: int) -> None:
        self._pickle_id = value
    def param(self, name: str, default: Any = NOTSET) -> DagParam:
        """
        Return a DagParam object for current dag.
        :param name: dag parameter name.
        :param default: fallback value for dag parameter.
        :return: DagParam instance for specified name and current dag.
        """
        return DagParam(current_dag=self, name=name, default=default)
    @property
    def tasks(self) -> list[Operator]: