### 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 (
            get_last_dagrun(
                self.dag_id, session=session, include_externally_triggered=include_externally_triggered
            )
            is not None
        )
    @property
    def dag_id(self) -> str:
        return self._dag_id
    @dag_id.setter
    def dag_id(self, value: str) -> None:
        self._dag_id = value
    @property
    def is_subdag(self) -> bool:
        return self.parent_dag is not None
    @property
    def full_filepath(self) -> str:
        """Full file path to the DAG.
        :meta private:
        """
        warnings.warn(
            "DAG.full_filepath is deprecated in favour of fileloc",
            RemovedInAirflow3Warning,
            stacklevel=2,
        )
        return self.fileloc
    @full_filepath.setter
    def full_filepath(self, value) -> None: