Yes. I realise that and it is helpful to think about different sizes of infinities. My favourite example that is easier to grok is the comparison between the set of natural numbers and the set of even numbers. That's like thinking from the inside - it works by looking at the members of the set. From the outside of the set infinite is infinite and since infinite is not a quantity then the set sizes are not comparable. Right now though I think we're talking about some finer points of theory that don't particularly matter outside of any particular application.
So, yes, we can use the intuition that although the number of points on a line segment are infinite, and the number of line segments in a square are also infinite, there are more points in a square. By extension the number of squares in a cube is infinite.... that leads to the number of points in shapes of various dimensions:
1D: inf (e.g. points on a line segment)
2D: inf^inf (points on a square)
3D: inf^inf^inf (points in a cube)
4D: inf^inf^inf^inf (points in a 4D hypercube)
etc....
If your definition is more restrictive for how the shapes go into the higher dimensions, such as it must be an exact slice of the higher dimensional shape and they must be packed without overlaps, then the density chnges somewhat. First what do I mean by an exact, non-overlapping slice, a row of points on a line segement. Then a row (or column) of equal length line segments that extend from edge to edge of the square... (extend into higher dimensions). Then the density of points increases proportional to the number of dimensions:
1D: inf
2D: inf^2
3D: inf^3
4D: inf^4
This is fun stuff btw :)