### 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)

"""Invocation instance of a DAG.
    A DAG run can be created by the scheduler (i.e. scheduled runs), or by an
    external trigger (i.e. manual runs).
    """
    __tablename__ = "dag_run"
    id = Column(Integer, primary_key=True)
    dag_id = Column(StringID(), nullable=False)
    queued_at = Column(UtcDateTime)
    execution_date = Column(UtcDateTime, default=timezone.utcnow, nullable=False)
    start_date = Column(UtcDateTime)
    end_date = Column(UtcDateTime)
    _state = Column("state", String(50), default=DagRunState.QUEUED)
    run_id = Column(StringID(), nullable=False)
    creating_job_id = Column(Integer)
    external_trigger = Column(Boolean, default=True)
    run_type = Column(String(50), nullable=False)
    conf = Column(PickleType)
    
    data_interval_start = Column(UtcDateTime)
    data_interval_end = Column(UtcDateTime)
    
    last_scheduling_decision = Column(UtcDateTime)
    dag_hash = Column(String(32))
    
    
    
    log_template_id = Column(
        Integer,