### 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)

from airflow.utils.state import DagRunState, State, TaskInstanceState
from airflow.utils.types import NOTSET, DagRunType
if TYPE_CHECKING:
    from datetime import datetime
    from sqlalchemy.orm import Query, Session
    from airflow.models.dag import DAG
    from airflow.models.operator import Operator
    from airflow.serialization.pydantic.dag_run import DagRunPydantic
    from airflow.serialization.pydantic.taskinstance import TaskInstancePydantic
    from airflow.typing_compat import Literal
    from airflow.utils.types import ArgNotSet
    CreatedTasks = TypeVar("CreatedTasks", Iterator["dict[str, Any]"], Iterator[TI])
    TaskCreator = Callable[[Operator, Iterable[int]], CreatedTasks]
RUN_ID_REGEX = r"^(?:manual|scheduled|dataset_triggered)__(?:\d{4}-\d{2}-\d{2}T\d{2}:\d{2}:\d{2}\+00:00)$"
class TISchedulingDecision(NamedTuple):
    """Type of return for DagRun.task_instance_scheduling_decisions."""
    tis: list[TI]
    schedulable_tis: list[TI]
    changed_tis: bool
    unfinished_tis: list[TI]
    finished_tis: list[TI]
def _creator_note(val):
    """Creator the ``note`` association proxy."""
    if isinstance(val, str):
        return DagRunNote(content=val)
    elif isinstance(val, dict):
        return DagRunNote(**val)
    else:
        return DagRunNote(*val)
class DagRun(Base, LoggingMixin):