### 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)

"""Stringified DAGs and operators contain exactly these fields."""
        if not cls.__serialized_fields:
            exclusion_list = {
                "parent_dag",
                "schedule_dataset_references",
                "task_outlet_dataset_references",
                "_old_context_manager_dags",
                "safe_dag_id",
                "last_loaded",
                "user_defined_filters",
                "user_defined_macros",
                "partial",
                "params",
                "_pickle_id",
                "_log",
                "task_dict",
                "template_searchpath",
                "sla_miss_callback",
                "on_success_callback",
                "on_failure_callback",
                "template_undefined",
                "jinja_environment_kwargs",
                
                "has_on_success_callback",
                "has_on_failure_callback",
                "auto_register",
                "fail_stop",
            }
            cls.__serialized_fields = frozenset(vars(DAG(dag_id="test"))) - exclusion_list