### 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)

"""A table for DAGs code.
    dag_code table contains code of DAG files synchronized by scheduler.
    For details on dag serialization see SerializedDagModel
    """
    __tablename__ = "dag_code"
    fileloc_hash = Column(BigInteger, nullable=False, primary_key=True, autoincrement=False)
    fileloc = Column(String(2000), nullable=False)
    
    last_updated = Column(UtcDateTime, nullable=False)
    source_code = Column(Text().with_variant(MEDIUMTEXT(), "mysql"), nullable=False)
    def __init__(self, full_filepath: str, source_code: str | None = None):
        self.fileloc = full_filepath
        self.fileloc_hash = DagCode.dag_fileloc_hash(self.fileloc)
        self.last_updated = timezone.utcnow()
        self.source_code = source_code or DagCode.code(self.fileloc)
    @provide_session
    def sync_to_db(self, session: Session = NEW_SESSION) -> None: