Let $R$ be a non-trivial ring with unity and $m\in\mathbf{N}^*$. Set $X:=(X_1,\ldots,X_m)$. For $\alpha\in\mathbf{N}^{[1,m]}$, let $X^\alpha$ denote the formal power series in $m$ indeterminates over $R$ such that $X^\alpha_\alpha=1$ and $X^\alpha_\beta=0$ for $\alpha\ne\beta$ ($\beta\in\mathbf{N}^{[1,m]}).$
What is the point of writing "$X:=(X_1,\ldots,X_m)$"? What set is $(X_1,\ldots, X_m)$ a member of? In an earlier section, the author defines $X$ as the formal power series in $1$ indeterminate for which $X_1=1$ and $X_n=0$ for $n\ne 1$. Are the $X_i$ values of that same power series $X$?
Not really. You have to understand $X_1, \dots X_m$ are just shortcuts for these series with coefficients indexed by $\mathbf N^m$: \begin{align} X_1&=\bigl(0,1_{(1,0,0,\dots,0)}, 0, 0,0,\dots \bigr) \\ X_2&=\bigl(0,1_{(0,1,0,\dots,0)}, 0, 0,0,\dots \bigr) \\ &\enspace\vdots \\ X_m &=\bigl(0,1_{(0,0,0,\dots,1)}, 0, 0,0,\dots \bigr) \end{align} and it can be shown that with Cauchy product and addition term by term, any series $(a_\underline \alpha)$, where $\underline\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)$, can be written explicitly as $$\sum_{\underline\alpha\in\mathbf N^m}c_{\underline\alpha}\underline X^{\underline\alpha}=\mkern-24mu\sum_{(\alpha_1,\alpha_2,\dots,\alpha_m)\in\mathbf N^m}\mkern-6muc_{(\alpha_1,\alpha_2,\dots,\alpha_m)} X_1^{\alpha_1} X_2^{\alpha_2}\dotsm X_m^{\alpha_m}.$$