### 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)

"""
        Retrieve the PID of the process processing the given file or None if the file is not being processed.
        :param file_path: the path to the file that's being processed.
        """
        if file_path in self._processors:
            return self._processors[file_path].pid
        return None
    def get_all_pids(self) -> list[int]:
        """
        Get all pids.
        :return: a list of the PIDs for the processors that are running
        """
        return [x.pid for x in self._processors.values()]
    def get_last_runtime(self, file_path) -> float | None:
        """
        Retrieve the last processing time of a specific path.
        :param file_path: the path to the file that was processed
        :return: the runtime (in seconds) of the process of the last run, or
            None if the file was never processed.
        """
        stat = self._file_stats.get(file_path)
        return stat.last_duration.total_seconds() if stat and stat.last_duration else None
    def get_last_dag_count(self, file_path) -> int | None: