The rapid integration of large language model-based conversational systems has intensified longstanding questions in mediated communication, human–computer interaction, and social presence research by placing language-generating systems within ongoing conversational exchanges. Rather than treating these systems only as information-retrieval tools, users often orient to them through interactional cues associated with responsiveness, continuity, and interlocutor recognition. However, existing analytical approaches do not adequately capture the multidimensional nature of these interactions. This study addresses this gap by proposing a computational framework grounded in communication theory, operationalized through three interaction indices: the Social Presence Index (SPI), the Social Bonding Index (SBI), and the Companion Communication Index (CCI). The framework is evaluated on more than 50,000 conversations from heterogeneous datasets (WildChat, LMSYS, DailyDialog, and MultiWOZ), enabling a comparative analysis between human–AI and human–human communication. Results show that human–human interactions exhibit higher social presence (SPI = 0.31 vs. 0.13), while human–AI interactions demonstrate greater structural continuity (CCI = 0.41 vs. 0.34), with statistically significant differences (p < 0.001). Segmentation analysis reveals non-linear interaction patterns, where conversational continuity peaks at intermediate levels of social presence in human–AI exchanges. These findings demonstrate that social presence, affective bonding, and conversational continuity are interdependent and context-sensitive, supporting a multidimensional framework for understanding and designing human–AI communication processes.