### 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)

test_dag_path = TEST_DAG_FOLDER / "test_scheduler_dags.py"
        async_mode = "sqlite" not in conf.get("database", "sql_alchemy_conn")
        log_file_loc = conf.get("logging", "DAG_PROCESSOR_MANAGER_LOG_LOCATION")
        try:
            os.remove(log_file_loc)
        except OSError:
            pass
        
        processor_agent = DagFileProcessorAgent(test_dag_path, 0, timedelta(days=365), [], False, async_mode)
        processor_agent.start()
        if not async_mode:
            processor_agent.run_single_parsing_loop()
        processor_agent._process.join()
        assert os.path.isfile(log_file_loc)