### 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)

"""
        Parse a given link, and verifies if it's a valid URL, or a 'mailto' link.
        Returns an iterator of invalid (owner, link) pairs.
        """
        for owner, link in self.owner_links.items():
            result = urlsplit(link)
            if result.scheme == "mailto":
                
                if not result.path:
                    yield result.path, link
            elif not result.scheme or not result.netloc:
                yield owner, link
class DagTag(Base):
    """A tag name per dag, to allow quick filtering in the DAG view."""
    __tablename__ = "dag_tag"
    name = Column(String(TAG_MAX_LEN), primary_key=True)
    dag_id = Column(
        StringID(),
        ForeignKey("dag.dag_id", name="dag_tag_dag_id_fkey", ondelete="CASCADE"),
        primary_key=True,
    )
    def __repr__(self):
        return self.name
class DagOwnerAttributes(Base):