### 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)

query.delete(synchronize_session="fetch")
        session.commit()
    def _log_file_processing_stats(self, known_file_paths):
        """
        Print out stats about how files are getting processed.
        :param known_file_paths: a list of file paths that may contain Airflow
            DAG definitions
        :return: None
        """
        
        
        
        
        
        
        
        
        headers = ["File Path", "PID", "Runtime", "# DAGs", "# Errors", "Last Runtime", "Last Run"]
        rows = []
        now = timezone.utcnow()
        for file_path in known_file_paths:
            last_runtime = self.get_last_runtime(file_path)
            num_dags = self.get_last_dag_count(file_path)
            num_errors = self.get_last_error_count(file_path)
            file_name = os.path.basename(file_path)
            file_name = os.path.splitext(file_name)[0].replace(os.sep, ".")
            processor_pid = self.get_pid(file_path)
            processor_start_time = self.get_start_time(file_path)
            runtime = (now - processor_start_time) if processor_start_time else None
            last_run = self.get_last_finish_time(file_path)