As large language models (LLMs) are increasingly integrated into emotionally sensitive domains, the structural integrity of their emotional intelligence (EI) becomes a critical frontier for safety and alignment. Current benchmarks often conflate superficial politeness with deep affective reasoning, failing to distinguish between perceptual accuracy and interactive efficacy. Here, we introduce FACET (Functional Affective Competence and Empathy Test), a psychometrically grounded framework comprising 480 expert-crafted items. Unlike previous metrics, FACET is theoretically anchored in the Mayer-Salovey-Caruso four-branch ability model, operationalizing EI through perception, facilitation, understanding, and management of emotions. Through an evaluation of nine frontier models (including GPT-5, Claude-Sonnet-4), we demonstrate that emotional intelligence is not a monolithic capability but is fragmented across cognitive and interactive dimensions. While frontier models demonstrate robust proficiency in objective emotion recognition and social reasoning, this does not consistently translate to interactive success. We categorize these discrepancies into three distinct performance profiles: cognitive-dominant, interactive-dominant, and context-dependent. These typologies indicate that emotional skills do not scale uniformly with general intelligence or model size; rather, they are shaped by specific alignment paradigms. Notably, we identify hidden emotion recognition as a universal performance bottleneck across all architectures. Our results suggest that current RLHF processes may optimize for "stochastic empathy", a statistical mimicry of emotional syntax, at the expense of integrated affective reasoning. These findings challenge the assumption of linear emotional scaling and provide a rigorous roadmap for developing socially aware agents capable of genuine clinical resonance.