### 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)

self._dag_ids = dag_ids
        self._async_mode = async_mode
        self._parsing_start_time: float | None = None
        self._dag_directory = dag_directory
        
        
        
        #
        
        
        if self._async_mode and self._direct_scheduler_conn is not None:
            os.set_blocking(self._direct_scheduler_conn.fileno(), False)
        self.standalone_dag_processor = conf.getboolean("scheduler", "standalone_dag_processor")
        self._parallelism = conf.getint("scheduler", "parsing_processes")
        if (
            conf.get_mandatory_value("database", "sql_alchemy_conn").startswith("sqlite")
            and self._parallelism > 1
        ):
            self.log.warning(
                "Because we cannot use more than 1 thread (parsing_processes = "
                "%d) when using sqlite. So we set parallelism to 1.",
                self._parallelism,
            )
            self._parallelism = 1
        
        self._file_process_interval = conf.getint("scheduler", "min_file_process_interval")
        
        
        self.print_stats_interval = conf.getint("scheduler", "print_stats_interval")