According to the following definition, a multi-interval is just a line segment for n=1, a square region for n=2, a cubic for n=3, and so on so forth, correct?
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Not necessarily a square region ($n=2$) and a cubic region ($n=3$), but rectangle and a box. For higher dimensions the word used seems to be hyperrectangle.
A multi-interval is often written as for example $[a_1, b_1] \times (a_2, b_2) \times \cdots \times (a_n, b_n]$.
md2perpe
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An old geometric term for it (without regard to co-ordinate axes) is $n$-dimensional orthogonal parallelipiped. – DanielWainfleet Sep 08 '18 at 20:24