Can closer points be considered more similar in T-SNE visualization?
I would present t-SNE as a smart probabilistic adaptation of the Locally-linear embedding. In both cases, we attempt to project points from a high dimensional space to a small one. This projection is done by optimizing the conservation of local distances (directly with LLE, preproducing a probabilistic distribution and optimizing the KL-divergence with t-SNE). Then if your question is, does it keep global distances, the answer is no. It will depend on the "shape" of your data (if the distribution is smooth, then distances should be somehow conserved). 

t-SNE actually doesn't work well on the swiss roll (your "S" 3D image) and you can see that, in the 2D result, the very middle yellow points are generally closer to the red ones than the blue ones (they are perfectly centered in the 3D image).  

An other good example of what t-SNE does is the clustering of handwritten digits. See the examples on this link:https://lvdmaaten.github.io/tsne/