Many-shot in-context learning (ICL) has emerged as a unique setup to both utilize and test the ability of large language models to handle long context. This paper delves into long-context language model (LCLM) evaluation through many-shot ICL. We first ask: what types of ICL tasks benefit from additional demonstrations, and how effective are they in evaluating LCLMs? We find that classification and summarization tasks show performance improvements with additional demonstrations, while translation and reasoning tasks do not exhibit clear trends. Next, we investigate the extent to which different tasks necessitate retrieval versus global context understanding. We develop metrics to categorize ICL tasks into two groups: (i) similar-sample learning (SSL): tasks where retrieval of the most similar examples is sufficient for good performance, and (ii) all-sample learning (ASL): tasks that necessitate a deeper comprehension of all examples in the prompt. Lastly, we introduce a new many-shot ICL benchmark, MANYICLBENCH, to characterize model's ability on both fronts and benchmark 12 LCLMs using MANYICLBENCH. We find that while state-of-the-art models demonstrate good performance up to 64k tokens in SSL tasks, many models experience significant performance drops at only 16k tokens in ASL tasks.