Are cartesian coordinates "more fundamental" than other coordinates, and are they inherently tied to $\mathbb R^n$? Are the Cartesian coordinates more "fundamental" than other coordinate systems?  When someone says $\mathbb R^n$ do we implicitly mean the set of points PLUS Cartesian coordinate system?  Sometimes I read "Cartesian space" for $\mathbb R^n$, but of course nobody is calling $\mathbb R^n$ "polar space" (You can see this in the Euclidean space wikipedia page).
This example seems to suggest to me that Cartesian coordinates are the "default" system on $\mathbb R^n$:
If you have a constant function $f=1$ from a subset $S$ of $\mathbb R^n$ into $\mathbb R$, and you do $\iint_{S}{1 \cdot dA}$, that is interpreted as the volume of a box with base area $S$ and height $1$.  This assumes that $S \subset \mathbb R^2$ is in Cartesian coordinates.
Another thing that suggests that Cartesian coordinates are more fundamental:
$\mathbb R^2$ is the "Cartesian product" of $\mathbb R$ with itself. 
 A: When I see a notation like $\mathbb{R}^2$ or $\mathbb{R} \times \mathbb{R}$, I tend to think of this as indicating an object considered as coming equipped with projection or coordinate functions $\pi_1: \mathbb{R}^2 \to \mathbb{R}$, $\pi_2: \mathbb{R}^2 \to \mathbb{R}$ which assign to each element of the domain a first and second coordinate. With regard to comments made under the question, note that any choice of linear basis of a 2-dimensional vector space $V$ can be used to produce such a suitable pair of projection functions (giving coordinates with respect to that basis), and one can go on to consider extra structure such as an inner product that makes those basis elements orthonormal elements, etc., so that one effectively derives a cartesian coordinate structure on $V$.  
The polar coordinate system is an entirely different creature, being based on a pair of maps $\mathbb{R}^2 \to \mathbb{R}_+$, $\mathbb{R}^2 \to \{angles\}$ that is not inherent in the notation (it's not even well-defined at the origin, or at least not so in any continuous way). In brief, I would say that unless the author explicitly declares some alternative coordinate system, you should think of cartesian coordinate systems (in the sense discussed above) as the default inherent within the very notation. 
