$\DeclareMathOperator\SL{SL}$The same argument (due to Siegel, in a classical form, of course), adelized, gives the analogous computation for any number field, and, yes, the corresponding Dedekind zeta appears. (To know that the adelic analogue computes the same thing, rather than the adelic quotient for $\SL_n$ being larger than the classical one, probably we invoke Strong Approximation, just to be safe.)
Certainly there is the issue of describing a reasonable normalization of the Haar measures. Of course, in a silly way, we could normalize them to get any result we wanted. For that matter, from a contemporary viewpoint, it is more stylish to define "Tamagawa measures" on semisimple groups $G$, designed so that the measure of $G_k\backslash G_{\mathbb A}$ is (often) $1$, or some other simple integer. With hindsight, Siegel's computation gives hints about how to do this. (See Weil's book "Adeles and algebraic groups".)
Operationally, from a neo-classical viewpoint, the important thing is to make natural choices of measures on $P_v$ and $K_v$, where $P_v$ is the $v$-adic minimal parabolic and $K_v$ is the $v$-adic maximal compact, which assemble to give a measure on $G_v$. It is natural to give $K_v$ measure $1$ at non-archimedean places. The measure on $P_v$ is made from a local multiplicative measure on $k_v^\times$, which has the natural normalization that $\mathfrak{o}_v^\times$ has measure $1$. At places $v$ not ramified over $\mathbb Q$, the additive measure naturally gives $\mathfrak{o}_v$ measure $1$. At absolutely ramified places, there is also a natural normalization as in Iwasawa–Tate theory, and this introduces a power of the discriminant, unsurprisingly.
At archimedean places $v$, thinking in terms of Iwasawa–Tate theory gives a natural measure on $P_v$. The chief ambiguity here is about $K_v$. One choice is to relate the measure to the natural measure of spheres. Another is to give $K_v$ measure $1$. And other choices that adapt to induction over the size $n$. These lead to different constants, attributible to these choices at archimedean places.
EDIT: in response to a question about why computing "the" volume of $\SL_n(\mathbb{Q})\backslash\SL_n({\mathbb A})$ should be essentially the same thing as that of $\SL_n(\mathbb Z)\backslash\SL_n(\mathbb R)$, and so on… The most typical (?) thing that is said is that (by Strong Approximation) $\SL_n(\mathbb A)=\SL_n(\mathbb Q)\SL_n(\mathbb{R})\prod_{p<\infty}\SL_n(\mathbb Z_p)$, from which
\begin{gather*}
\SL_n(\mathbb Q)\backslash\SL_n(\mathbb A)/\prod_{p<\infty}\SL_n(\mathbb Z_p)
\approx (\SL_n(\mathbb Q)\cap \SL_n(\mathbb R)\prod_{p<\infty}\SL_n(\mathbb Z_p))\backslash\SL_n(\mathbb R) \\
\;=\; \SL_n(\mathbb Z)\backslash\SL_n(\mathbb R).
\end{gather*}
In particular, a right $\SL_n(\mathbb R)$-invariant measure on one gives the same on the other. The measures on the various $X_N$ can be normalized to make everything consonant… A possibly disappointing aspect of this viewpoint is that it gives no origin story, despite being an orthodox viewpoint.
Another viewpoint, which in the long term may be more explanatory, is in terms of projective limits of classical quotients $X_N=\Gamma_N\backslash\SL_n(\mathbb R)$, where $\Gamma_N$ is the principal congruence subgroup of level $N$. When $M$ divides $N$, there is the natural surjection $X_N\to X_M$. Ordering positive integers by divisibility, the projective limit (a type of non-abelian solenoid) is naturally an object of interest. Already we could look at the proj lim of circles $\mathbb R/N\mathbb Z$, or even the proj lim of circles $\mathbb R/2^n\mathbb Z$. These solenoids were studied from a topological viewpoint in the early 1940s by Eilenberg and MacLane. The latter case gives $(\mathbb R\times \mathbb Q_2)/\mathbb Z[{1\over 2}]^\Delta$, where $\mathbb Q_2$ is the $2$-adic numbers. An interesting operational point is that $\mathbb Q_2$ acts on the family of circles $\mathbb R/2^n\mathbb Z$, although it does not act on the individuals.
Similarly the proj lim of $\mathbb R/N\mathbb Z$ is $\mathbb A/\mathbb Q$. That is, in fact every $\mathbb Q_p$ acts on that family of circles, though not on individuals.
Similarly, $\SL_n(\mathbb Q_p)$ acts on the projective family of classical arithmetic quotients $X_N$, though not on individuals.
(The emergence of adelic things in some areas of physics also seems to be not just by fiat of definition, but because one takes a projective limit of "charge lattices", whatever those are. :) I don't know anything beyond those words…)
