Note that this question has already been asked and answered on Mathematics SE. I think it is a really good answer, and I am not sure I can add much to it. So I will just summarize it here for future reference.
Essentially areas and volumes cannot always be treated on equal footing. Areas and volumes scale differently. For the cone there is proportionally more volume located near the base of the solid compared to how much area is proportionally near the base of the triangle. This is why the center of mass of the cone is located closer to the base than it is for the triangle.
To relate this more to physics, and to add some new information, I will say that this issue comes up in other places as well. For example, you could imagine a thin disk as just a bunch of thin rods aligned so that they are all concentric. Then one could wonder why the moment of inertia of a thin rod of mass $M$ and length $L$ about its center is $\frac13ML^2=\frac43M(L/2)^2$ but the moment of inertia of a thin disk of mass $m$ and radius $r$ is $\frac12mr^2$. You would expect both of the constant factors out front to be the same, but this is not what happens. (However, you are free to think of the disk as a bunch of really small wedges. But in this case you are not moving between the "number of dimensions" since the wedges and the disk are both areas).
I suppose this goes to show that there are differences between countably and uncountably infinite sets of objects. In order to move from the triangle to the cone or from the rod to the cylinder you have to give the smaller elements some thickness in the new dimension you are considering. This moves you from a countable set of objects to an uncountable set of objects. Hence we get different results.