### 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)

max_active_runs = Column(Integer, nullable=True)
    has_task_concurrency_limits = Column(Boolean, nullable=False)
    has_import_errors = Column(Boolean(), default=False, server_default="0")
    
    next_dagrun = Column(UtcDateTime)
    
    next_dagrun_data_interval_start = Column(UtcDateTime)
    next_dagrun_data_interval_end = Column(UtcDateTime)
    
    next_dagrun_create_after = Column(UtcDateTime)
    __table_args__ = (
        Index("idx_root_dag_id", root_dag_id, unique=False),
        Index("idx_next_dagrun_create_after", next_dagrun_create_after, unique=False),
    )
    parent_dag = relationship(
        "DagModel", remote_side=[dag_id], primaryjoin=root_dag_id == dag_id, foreign_keys=[root_dag_id]
    )
    schedule_dataset_references = relationship(
        "DagScheduleDatasetReference",
        cascade="all, delete, delete-orphan",
    )
    schedule_datasets = association_proxy("schedule_dataset_references", "dataset")
    task_outlet_dataset_references = relationship(
        "TaskOutletDatasetReference",
        cascade="all, delete, delete-orphan",
    )
    NUM_DAGS_PER_DAGRUN_QUERY = airflow_conf.getint(
        "scheduler", "max_dagruns_to_create_per_loop", fallback=10
    )
    def __init__(self, concurrency=None, **kwargs):