As conversational user interfaces (CUIs) evolve, trust and identity challenges become increasingly pronounced. The identities of CUIs are multifaceted, incorporating individual names like Siri or Alexa, the companies behind them like Apple or Amazon, and various attributes, e.g., race, gender, and class, as perceived by people and/or as designed into these CUIs. Identity is also encoded in the embodiment, be it as an abstract animation on a watch, an avatar in virtual reality, or a humanoid robot, as well as in the backstory designers give these agents. But, if identity is fragmented, e.g., across multiple physical forms, if and how users can establish trust becomes difficult to address. Drawing from diverse fields including ethics, design, and engineering, we explore the hurdles posed by ambiguous identities. A dynamic embodiment of a CUI across multiple devices presents technical complexities, and importantly, it raises ethical dilemmas surrounding trust. In this workshop, we aim to synthesize research goals and methods to further probe the intricacies of identity fragmentation and its implications for user trust in CUIs. To pursue a collaborative debate, we formulate that trust and identity suffer from the chicken or egg dilemma; should issues surrounding identity be resolved first before trust can even be conceived to be possible between humans and CUIs? Can users truly trust a CUI that lacks a consistent and transparent identity, and would that trust be different for different embodiments and platforms? We consider that trust itself perhaps should be questioned given that the issues surrounding identity are not resolved. We additionally discuss whether a uniform identity across all interfaces is conducive to user trust, or whether the adoption of distinct personas on disparate platforms is more effective in engendering user trust.