### Apache Airflow version

Other Airflow 2 version (please specify below)

### What happened

Greetings!
Recently I’ve faced a problem. It seems that assigning object, which can’t be serialized to JSON, to the dag_run.conf dict cause critical errors with UI.
After executing code example in "How to reproduce": 
Grid representation of the DAG breaks with following result:
<img width="800" alt="Pasted Graphic" src="https://github.com/apache/airflow/assets/79107237/872ccde5-500f-4484-a36c-dce6b7112286">
Browse -> DAG Runs also becomes unavailable.
<img width="800" alt="Pasted Graphic 1" src="https://github.com/apache/airflow/assets/79107237/b4e3df0c-5324-41dd-96f3-032e706ab7a9">
Dag itself continues to work correctly, this affects only UI graph and dagrun/list/
I suggest to use custom dictionary with restriction on setting non json values.

### What you think should happen instead

Raise an error

### How to reproduce

Execute following task. 
Composer 2 version 2.4.2
Airflow version 2.5.3
```
@task 
def test_task(**context):
    context['dag_run'].conf["test"] = np.int64(1234)
```

### Operating System

Ubuntu 20.04.6 LTS

### Versions of Apache Airflow Providers

_No response_

### Deployment

Google Cloud Composer

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [X] 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)

"""For storage of arbitrary notes concerning the dagrun instance."""
    __tablename__ = "dag_run_note"
    user_id = Column(
        Integer,
        ForeignKey("ab_user.id", name="dag_run_note_user_fkey"),
        nullable=True,
    )
    dag_run_id = Column(Integer, primary_key=True, nullable=False)
    content = Column(String(1000).with_variant(Text(1000), "mysql"))
    created_at = Column(UtcDateTime, default=timezone.utcnow, nullable=False)
    updated_at = Column(UtcDateTime, default=timezone.utcnow, onupdate=timezone.utcnow, nullable=False)
    dag_run = relationship("DagRun", back_populates="dag_run_note")
    __table_args__ = (
        PrimaryKeyConstraint("dag_run_id", name="dag_run_note_pkey"),
        ForeignKeyConstraint(
            (dag_run_id,),
            ["dag_run.id"],
            name="dag_run_note_dr_fkey",
            ondelete="CASCADE",
        ),
    )
    def __init__(self, content, user_id=None):
        self.content = content
        self.user_id = user_id
    def __repr__(self):
        prefix = f"<{self.__class__.__name__}: {self.dag_id}.{self.dagrun_id} {self.run_id}"
        if self.map_index != -1:
            prefix += f" map_index={self.map_index}"
        return prefix + ">"