Wikipedia defines the Real Coordinate Space as follows:
For any natural number $n$, the set $\mathbb{R}^n$ consists of all n-tuples of real numbers ($\mathbb{R}$). It is called (the) $n$-dimensional real space
Now, let $C_i, 1 ≤ i ≤ n$, be $n$ sets. Wikipedia defines a coordinate space of $n$ dimensions as a set $S$ together with a surjective partial mapping $$ \varphi : C_1 \times \dots \times C_n \to S $$
If I understand it correctly, converting one Coordinate system into another, e.g. Cartesian coordinates into Spherical coordinates is just using sets that are different to $C_i, 1 ≤ i ≤ n$ together with a different mapping $\varphi$. However, both coordinate systems still index the same set of elements e.g. the elements from $S$ or the elements / tupels from $\mathbb{R}^n$
Q: Is my assumption that two different coordinate system can refere to the same set of elements correct or am I misunderstanding something?
Yes, even if you work with one space, a coordinating system could vary.
$\mathbb{R}^n$ as a real vector space it is already parametrized by itself, $\phi:\mathbb{R}^n \to \mathbb{R}^n$ is identity and therefore could be omitted. On the other hand, when more general vector space $V$ of dimension $n$ is given, then coordinate system that is also linear isomorphism is substantial: $\phi:\mathbb{R}^n\to V$. Note, that any $n\times n$ invertible matrix $M$ yields new coordinate system $\phi \circ M:\mathbb{R}^n\to V$ that is also an isomorphism.
In calculus switch between polar and cartesian coordinate system can help with solving integral. Interesting example is Gaussian integral: $\int e^{-x^2}dx$. Indefinite integral can not be expessed in terms of elementary functions, but $\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt\pi$ and the proof uses polar coordinates.
Both spherical and cylindrical coordinate spaces describe the same set: $\mathbb{R}^3$.
By these examples I've tried to compensate vague wikipedia article, that you have mentioned. With definition that broad nearly everything is coordinate space: permutation of a set, group epimorphism, projection, n-ary relation, graph coloring etc. From my experience "coordinates" always means "elements of $\mathbb{R}^n$ (or $\mathbb{R}$ in particular)", but it's just an opinion.