Clearly we can't have three collinear points on a sphere. Look
at the "on a sphere" case and assume there is a configuration
where they do not all lie on a single sphere. Any three points
define a plane. The locus of points equidistant from these three
is a line perpendicular to the plane (through the circumcentre).
There are at most two points on such a line which are unit
distance from the original three, $P$ and $Q$ say. These are the
centres of two unit circles, and one of the remaining two points
must lie on each sphere. Note that $P$ and $Q$ are related by a
reflection in the original plane. There are five sets of four
points in the original configuration. Each set defines a unit
sphere, and if two spheres are the same, then all are. So there
are five spheres and the centres are related by reflections in
the planes defined by triangles. [From here is a bathtime
intuition which, per comments doesn't work. However note that
each pair of centres is related by a reflection, which may not
map other centres t