Folding a square to produce a torus has an analogy with "folding" a cube to produce what? Is there a name for the resulting 4D structure?
Yes. Perhaps unfortunately, such an object is also called a torus. If you want to be precise, you might use the word "hypertorus" or say explicitly "$n$-dimensional torus". In your case, you would be interested in the $3$-torus embedded in $4$D.
In general, if we write $S^1$ for the circle, then $S^1 \times S^1 \times \cdots \times S^1$ is called a torus (by analogy with the classical case $S^1 \times S^1$).
In particular, a square is $[0,1] \times [0,1]$. So when we glue together the boundaries, we're glueing the $[0,1]$s into circles. So we wind up with $S^1 \times S^1$.
What about a cube? It's $[0,1] \times [0,1] \times [0,1]$. Again, we glue opposite sides of the cube together, and we wind up with $S^1 \times S^1 \times S^1$.
I hope this helps ^_^
Such an object is also called a genus-3 handlebody.