In numerical optimization, the characterization of optimization problems and their properties has been a long-standing issue. Overcoming it is a crucial prerequisite for many optimization-related tasks such as building quality benchmarks, algorithm selection, and algorithm configuration. Several approaches to extracting features from single-objective optimization problems have been proposed but they all have some inherent limitations and thus offer an incomplete look at the problems and their properties. In this work, we extend and improve our previous work on existing topological features that offer a new look at optimization problems where their similarity is quantified in terms of the appearance of topological structures. We show that topologically inspired features are not correlated with existing state-of-the-art landscape feature groups, meaning that they capture different and thus complementary information. Topological features are subsequently used to classify problem instances from the COmparing Continuous Optimizers (COCO) benchmark showing that similar problems most often have similar topological features. Further, we perform a sensitivity analysis of the proposed methodology to its hyperparameters and provide some additional insight into the behavior of the topological features. Lastly, we also investigate topological features and their generalization across different sample sizes.