The discussion in the comments has triggered my curiosity: I've done a little bit of research and I almost immediately stumbled upon the Wikipedia entry on the Peano-Jordan measure. There I found exactly the terms "simple sets" and "polyrectangles" used as synonyms for the unions of finite families of $n$-rectangles, i.e. for the set $$ S=\bigcup_{i=1}^q C_i\quad q\in\Bbb N_{>0} $$ where $n$ is the dimension of the Euclidean $\Bbb R^n$ considered and $C_i$ are $n$-rectangles defined as $C_i=[a_{1i}, b_{1i})\times\cdots\times[a_{ni}, b_{ni})$. However, I was not able to find where these sets are called in this way within the references stated in the relevant section of the entry: thus I started searching in my personal library. I found the locution "plurirectangles" (plurirettangoli in Italian) in the classical monograph [1], chapter II, §2.7, p. 119, as shown in the picture below:
This makes me sufficiently confident in proposing the adoption of this, somewhat traditional, terminology (i.e. plurirectangle, see the discussion in the comments by Pietro Majer and Daniele Tampieri).
Reference
[1] Federico Cafiero (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.