Analogical reasoning is at the core of human cognition, serving as an important foundation for a variety of intellectual activities. While prior work has shown that LLMs can represent task patterns and surface-level concepts, it remains unclear whether these models can encode high-level relational concepts and apply them to novel situations through structured comparisons. In this work, we explore this fundamental aspect using proportional and story analogies, and identify three key findings. First, LLMs effectively encode the underlying relationships between analogous entities; both attributive and relational information propagate through mid-upper layers in correct cases, whereas reasoning failures reflect missing relational information within these layers. Second, unlike humans, LLMs often struggle not only when relational information is missing, but also when attempting to apply it to new entities. In such cases, strategically patching hidden representations at critical token positions can facilitate information transfer to a certain extent. Lastly, successful analogical reasoning in LLMs is marked by strong structural alignment between analogous situations, whereas failures often reflect degraded or misplaced alignment. Overall, our findings reveal that LLMs exhibit emerging but limited capabilities in encoding and applying high-level relational concepts, highlighting both parallels and gaps with human cognition.