Self-disclosure, the sharing of one's thoughts and feelings, is affected by the perceived relationship between individuals. While chatbots are increasingly used for self-disclosure, the impact of a chatbot's framing on users' self-disclosure remains under-explored. We investigated how a chatbot's description of its relationship with users, particularly in terms of ephemerality, affects self-disclosure. Specifically, we compared a Familiar chatbot, presenting itself as a companion remembering past interactions, with a Stranger chatbot, presenting itself as a new, unacquainted entity in each conversation. In a mixed factorial design, participants engaged with either the Familiar or Stranger chatbot in two sessions across two days, with one conversation focusing on Emotional- and another Factual-disclosure. When Emotional-disclosure was sought in the first chatting session, Stranger-condition participants felt more comfortable self-disclosing. However, when Factual-disclosure was sought first, these differences were replaced by more enjoyment among Familiar-condition participants. Qualitative findings showed Stranger afforded anonymity and reduced judgement, whereas Familiar sometimes felt intrusive unless rapport was built via low-risk Factual-disclosure.