### 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)

return f"<DAG: {self.dag_id}>"
    def __eq__(self, other):
        if type(self) == type(other):
            
            
            return all(getattr(self, c, None) == getattr(other, c, None) for c in self._comps)
        return False
    def __ne__(self, other):
        return not self == other
    def __lt__(self, other):
        return self.dag_id < other.dag_id
    def __hash__(self):
        hash_components = [type(self)]
        for c in self._comps:
            
            if c == "task_ids":
                val = tuple(self.task_dict)
            else:
                val = getattr(self, c, None)
            try:
                hash(val)
                hash_components.append(val)
            except TypeError:
                hash_components.append(repr(val))
        return hash(tuple(hash_components))
    
    def __enter__(self):
        DagContext.push_context_managed_dag(self)
        return self
    def __exit__(self, _type, _value, _tb):