### Apache Airflow version

Other Airflow 2 version (please specify below)

### What happened

Airflow 2.3.3

Raising the `AirflowSkipException` in the source code, using the `DockerOperator`, is supposed to mark the task as skipped, according to the [docs](https://airflow.apache.org/docs/apache-airflow/stable/concepts/tasks.html#special-exceptions). However, what happens is that the task is marked as failed with the logs showing `ERROR - Task failed with exception`.

### What you think should happen instead

Tasks should be marked as skipped, not failed.

### How to reproduce

Raise the `AirflowSkipException` in the python source code, while using the `DockerOperator`.

### Operating System

Ubuntu 20.04.5 LTS (GNU/Linux 5.4.0-125-generic x86_64)

### Versions of Apache Airflow Providers

_No response_

### Deployment

Docker-Compose

### 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)

]
        stringio_patcher = mock.patch("airflow.providers.docker.operators.docker.StringIO")
        stringio_mock = stringio_patcher.start()
        stringio_mock.side_effect = lambda *args: args[0]
        operator = DockerOperator(
            api_version="1.19",
            command="env",
            environment={"UNIT": "TEST"},
            private_environment={"PRIVATE": "MESSAGE"},
            env_file="ENV=FILE\nVAR=VALUE",
            image=TEST_IMAGE,
            network_mode="bridge",
            owner="unittest",
            task_id="unittest",
            mounts=[Mount(source="/host/path", target="/container/path", type="bind")],
            mount_tmp_dir=True,
            entrypoint=TEST_ENTRYPOINT,
            working_dir="/container/path",
            shm_size=1000,
            host_tmp_dir=TEST_HOST_TEMP_DIRECTORY,
            tmp_dir=TEST_AIRFLOW_TEMP_DIRECTORY,
            container_name="test_container",
            tty=True,
        )
        caplog.clear()
        with caplog.at_level(logging.WARNING, logger=operator.log.name):
            operator.execute(None)
            warning_message = (
                "Using remote engine or docker-in-docker and mounting temporary volume from host "
                "is not supported. Falling back to `mount_tmp_dir=False` mode. "