### 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)

return cls.__serialized_fields
    def get_edge_info(self, upstream_task_id: str, downstream_task_id: str) -> EdgeInfoType:
        """Return edge information for the given pair of tasks or an empty edge if there is no information."""
        
        empty = cast(EdgeInfoType, {})
        if self.edge_info:
            return self.edge_info.get(upstream_task_id, {}).get(downstream_task_id, empty)
        else:
            return empty
    def set_edge_info(self, upstream_task_id: str, downstream_task_id: str, info: EdgeInfoType):
        """
        Set the given edge information on the DAG.
        Note that this will overwrite, rather than merge with, existing info.
        """
        self.edge_info.setdefault(upstream_task_id, {})[downstream_task_id] = info
    def validate_schedule_and_params(self):
        """
        Validate Param values when the schedule_interval is not None.
        Raise exception if there are any Params in the DAG which neither have a default value nor
        have the null in schema['type'] list, but the DAG have a schedule_interval which is not None.
        """
        if not self.timetable.can_be_scheduled:
            return
        for k, v in self.params.items():
            
            if not v.has_value and ("type" not in v.schema or "null" not in v.schema["type"]):
                raise AirflowException(
                    "DAG Schedule must be None, if there are any required params without default values"
                )
    def iter_invalid_owner_links(self) -> Iterator[tuple[str, str]]: