Category-level object pose and shape estimation from a single depth image has recently drawn research attention due to its potential utility for tasks such as robotics manipulation. The task is particularly challenging because the three unknowns, object pose, object shape, and model-to-measurement correspondences, are compounded together, but only a single view of depth measurements is provided. Most of the prior work heavily relies on data-driven approaches to obtain solutions to at least one of the unknowns, and typically two, risking generalization failures if not designed and trained carefully. The shape representations used in the prior work also mainly focus on point clouds and signed distance fields (SDFs). In stark contrast to the prior work, we approach the problem using an iterative estimation method that does not require learning from pose-annotated data. Moreover, we construct and adopt a novel mesh-based object active shape model (ASM), which additionally maintains vertex connectivity compared to the commonly used point-based object ASM. Our algorithm, ShapeICP, is based on the iterative closest point (ICP) algorithm but is equipped with additional features for the category-level pose and shape estimation task. Although not using pose-annotated data, ShapeICP surpasses many data-driven approaches that rely on pose data for training, opening up a new solution space for researchers to consider.