Points lying in a common line are called colinear. Points lying in common plane are called coplanar. Points lying in a common (3-dimensional) space are called cospatial. But how would you call points lying in a common $n$-dimensional space? Can you just call them "co-$n$-spatial"?
Is really that meaning of cospatial generally agreed on? I don't think so. And I don't think there's a generally agreed on term for generalizations of the concept.
However some constructs would indicate the meaning of words from it's parts. So co-linear indicates they are on the same line, co-planar indicates that they are on the same plane.
Generalizing this construct co-spatial would mean they are in the same space (but remember that space is often used in a more general sense than just 3-dimensional Euclidean spaces).
On the other hand if we want to be more specific we can be more specific and say 3-space or if it's a subspace talk about 3-subspace. The term that conveys that would be co-3-spatial or co-3-subspatial. The special case where the subspace is one dimension less than the containing space it would be co-hyperplanar.
However as these are not generally agreed on terms you should probably define them somewhere, at least unless they are uniquely defined by the context. Choosing a good name is useful however as it would be easier to remember what it is.