### 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)

if "params" in self.default_args:
            params.update(self.default_args["params"])
            del self.default_args["params"]
        
        self.params = ParamsDict(params)
        if full_filepath:
            warnings.warn(
                "Passing full_filepath to DAG() is deprecated and has no effect",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
        validate_key(dag_id)
        self._dag_id = dag_id
        if concurrency:
            
            warnings.warn(
                "The 'concurrency' parameter is deprecated. Please use 'max_active_tasks'.",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
            max_active_tasks = concurrency
        self._max_active_tasks = max_active_tasks
        self._pickle_id: int | None = None
        self._description = description
        
        back = sys._getframe().f_back
        self.fileloc = back.f_code.co_filename if back else ""
        self.task_dict: dict[str, Operator] = {}
        
        tz = None