Online machine learning (ML) inference services provide users with an interactive way to request for predictions in real-time. To meet the notable computational requirements of such services, they are increasingly being deployed in the Cloud. In this context, the efficient provisioning and optimization of ML inference services in the Cloud is critical to achieve the required performance and meet the dynamic queries by end-users. Existing provisioning solutions focus on framework parameter tuning and infrastructure resources scaling, without considering deployments based on containerization technologies. The latter promises reproducibility and portability features for ML inferences services. There is limited knowledge about the impact of distinct deployment schemes at the container-level on the performance of online ML inference services, particularly on how to exploit multi-container deployments and its relation with processor and memory affinity. In light of this, in this paper we investigate experimentally the containerization of ML inference services and analyze the performance of multi-container deployments that partition the threads belonging to an online learning application into multiple containers in each node. This paper shares the findings and lessons learned from conducting realistic client patterns on an image classification model across numerous deployment configurations, especially including the impact of container granularity and its potential to exploit processor and memory affinity. Our results indicate that finegrained multi-container deployments and affinity are useful for improving performance (both throughput and latency). In particular, our experiments on single-node and four-node clusters show up to 69% and 87% performance improvement compared to the single-container deployment, respectively.