Relative consistency of $\mathsf{ZFC}-$Ext$+\neg$Ext+"every set has a unique powerset"

As Noah commented. ZFC + Ur-elements is known to be consistent relative to ZFC. It is usually referred to by "ZFA", i.e. ZF with atoms. In one formulation of that theory ONLY empty objects violate Extensionality! So we do have many empty objects (the Ur-elements). However all non-empty sets are extensional! That is, "No distinct non-empty sets are co-extensional (i.e. have the same elements)". Clearly this theory satisfy all of your conditions.

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