Complex data structures are now prevalent in biological applications. Statistical analysis of such data requires sophisticated representations and distances that can be used for their comparison and subsequent modeling. To this end, we propose a hybrid distance for classification of complex biological data objects that is able to capture shape variation of disparate components making up the objects. In particular, the hybrid distance quantifies shape differences in a curve component and topo-geometric differences in a point cloud component composing the complex object under study. For the curve component, we leverage a shape distance originating from the elastic shape analysis framework, which allows for computationally efficient and natural quantification of shape differences. For the point cloud component, we use the bottleneck distance between persistence diagrams, which capture birth and death times of topological features in the point clouds; as such, this distance captures complementary shape features of the data objects under study. To define the hybrid distance, the two individual distances are integrated with an appropriate weight parameter that allows the user to control their contribution in distance-based statistical tasks such as classification or clustering. We demonstrate the utility of the proposed framework for classification using simulations and two real data scenarios: Hawaiian Drosophila fly wings and pyramidal neuron trees.