For any domain $R$ an element of $R[[X]]$ is invertible if and only if the constant term is invertible in $R$.

Applying this repeatedly, one gets that an element of $\mathbb{Z}[[X_1,\dots,X_n]]$ is invertible (in this domain) if and only if its constant term is invertible in $\mathbb{Z}$ thar is it is in $\pm 1 + (X_1,\dots,X_n)$.

So, regarding your 'in particular' with finitely many variables, you have for $S$ the intersection of $\mathbb{Q}(X_1,\dots,X_n)$ and $\mathbb{Z}[[X_1,\dots,X_n]]$ that is the elements are a quotient of two rational (or equivalently integral) polynomials that are an integral power series, as in the question, that $S^{\ast}$ is contained in the set of power series with constant coeffiecient $\pm 1$ as$S^{\ast} \subset \mathbb{Z}[[X_1,\dots,X_n]]^{\ast}$. Yet, not each power series with constant coefficient $\pm 1$ is an element of $S^{\ast}$ for example as $\mathbb{Q}(X_1,\dots,X_n)$ is countable while there are uncountably many power series with constant coefficient $\pm 1$.

Thus, using the notation of the question, $S^{\ast} \subset \pm 1 + (X_1,\dots,X_n)$ yet the inclusion is strict, and this is not an equality.

The case of infinitely many variables: since you consider quotients of polynomials one can reduce to only considering the substructure where only the (finitely many) variables occuring in the polynomials are present (for each quotient individually).

Alternatively, we mainly need that the constant term of an invertible power series (finitely or infinitely many variable) is invertible in the base domain. This follows just by noting that the constant term of the product is the product of the constant terms, so if the product is $1$ it/they have to be invertible. Thus also showing the inclusion. That it is strict follows by restricting to a substructure with finitely many variables (or a suitable adaption of the cardinality argument).

One thing I am now somehow not completely sure about, though I think so, (but in any case it is not needed here) is whether in infinitely many variables also the invertability of the constant term is sufficient to imply that the power series is invertible.