### 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 last start time for processing a specific path.
        :param file_path: the path to the file that's being processed
        :return: the start time of the process that's processing the
            specified file or None if the file is not currently being processed.
        """
        if file_path in self._processors:
            return self._processors[file_path].start_time
        return None
    def get_run_count(self, file_path) -> int:
        """
        Return the number of times the given file has been parsed.
        :param file_path: the path to the file that's being processed.
        """
        stat = self._file_stats.get(file_path)
        return stat.run_count if stat else 0
    def get_dag_directory(self) -> str:
        """Return the dag_director as a string."""
        if isinstance(self._dag_directory, Path):
            return str(self._dag_directory.resolve())
        else:
            return str(self._dag_directory)
    def set_file_paths(self, new_file_paths):