How does the usual integration manipulations make sense here? 
In my book (which uses Cartesian coordinate system), it is written for any volume $V$:
$\displaystyle \iiint_{V} f(x,y,z)\ dxdydz + \iiint_{V} g(x,y,z)\ dxdydz=\iiint_{V} [f(x,y,z)+g(x,y,z)]\ dxdydz \tag1$

Considering Cartesian coordinate system,
it may be possible that whatever limits we put in the limits of integration, we cannot construct certain shapes $V^*$
For such shapes, how can the following manipulations make sense?:
$\displaystyle \iiint_{V^*} f(x,y,z)\ dxdydz + \iiint_{V^*} g(x,y,z)\ dxdydz=\iiint_{V^*} [f(x,y,z)+g(x,y,z)]\ dxdydz \tag2$
as all the terms are meaningless.
 A: As @saulspatz already pointed out in the comments: There is a difference between multiple integrals and iterated integrals. 
The term multiple integral refers to integration over multidimensional shapes. Multiple integrals are defined by splitting the shape you want to integrate over into pieces and by building a Riemann sum out of these pieces. If the Riemann sum converges then the multiple integral exists. (In 3D you can think of splitting some volume into small cubes and adding them up, for more details you could start here https://en.wikipedia.org/wiki/Multiple_integral#Mathematical_definition).
The term iterated integral is used to refer to nested 1D integrals. An iterated integral is usually computed by solving each 1D integral on it's own, starting at the innermost integral. 
What connects these two terms? There are important results in calculus which tell you how you can calculate some multiple integrals by transforming them into iterated integrals (for example Fubini's theorem). 
Just because you cannot come up with a way to write a multiple integral as iterated integral does not mean that the multiple integral you wrote down is meaningless. Or in other words: The definition of multiple integrals does not use iterated integrals. They are two different concepts but it is often very handy to use iterated integrals to compute multiple integrals.
