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We in mathematics say (so if you'll have to use, what will sound natural to you "on/in"?):
Two objects are coplanar if they both lie in the same plane.
Or
Two objects are coplanar if they both lie on the same plane.
So what sounds natural: "on" or "in" or both of them sound equally natural?
Also, is there any AmE-BrE difference?
Both are completely natural, but imply different ways of looking at the spatial orientation of the elements.
"In the same plane" is probably the more accurate from the mathematical perspective, since a plane has no depth. A point cannot rest "on" the plane the way a cup would rest on a table, since that would mean it is above the plane and therefore not part of the plane.
However, "on the same plane" is a common expression. Even though they should know it represents a mathematical fallacy, mathematicians persist in using the same language to describe these spatial relationships that they would use for real-world objects.
So, use whichever sounds better to you.
Speaking of a flat object (in geometry), we assume that all its points belong to some plane or lie in that plane:
Given a line in a plane, there exists at least one point in the plane that is not on the line. If two points lie in a plane, the line containing them lies in the plane. https://en.wikipedia.org/wiki/Point%E2%80%93line%E2%80%93plane_postulate
If that plane is the same for two such objects, those two are coplanar, so sentence 1 is correct. It's different in case of putting something on a flat list of some material (consider e. g. something shown in a picture vs a fly on that picture).
