### 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)

"""Add a formatted logger to the task instance.
            This allows all logs to surface to the command line, instead of into
            a task file. Since this is a local test run, it is much better for
            the user to see logs in the command line, rather than needing to
            search for a log file.
            :param ti: The task instance that will receive a logger.
            """
            format = logging.Formatter("[%(asctime)s] {%(filename)s:%(lineno)d} %(levelname)s - %(message)s")
            handler = logging.StreamHandler(sys.stdout)
            handler.level = logging.INFO
            handler.setFormatter(format)
            
            if not any(isinstance(h, logging.StreamHandler) for h in ti.log.handlers):
                self.log.debug("Adding Streamhandler to taskinstance %s", ti.task_id)
                ti.log.addHandler(handler)
        if conn_file_path or variable_file_path:
            local_secrets = LocalFilesystemBackend(
                variables_file_path=variable_file_path, connections_file_path=conn_file_path
            )
            secrets_backend_list.insert(0, local_secrets)
        execution_date = execution_date or timezone.utcnow()
        self.validate()
        self.log.debug("Clearing existing task instances for execution date %s", execution_date)
        self.clear(
            start_date=execution_date,
            end_date=execution_date,