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There cannot be a "global" result in complete generality, but there certainly can be results that are "local" over $S$. Here is one global counterexample. Let $S$ be $\mathbb{P}^1$. Let $Z$ be $S\times S$, and let $\Delta:S \to Z$ be the diagonal morphism. Let $(\pi:X\to Z,\chi:Z\to X)$ be the vector bundle of rank $1$ such that $\chi^*\mathcal{O}_X(\chi(Z))$ is isomorphic to $\text{pr}_1^*\mathcal{O}_S(0)\otimes \text{pr}_2^*\mathcal{O}_S(2)$ as an invertible sheaf on $Z=S\times S$. Similarly, let $(\rho:Y\to Z,\upsilon:Z\to Y)$ be the vector bundle of rank $1$ such that $\upsilon^*\mathcal{O}_Y(\upsilon(Z))$ is isomorphic to $\text{pr}_1^*\mathcal{O}_S(1)\otimes \text{pr}_2^*\mathcal{O}_S(1)$.

Edit. I realized that there is an easier example underlying the example above. Again let $S$ be $\mathbb{P}^1$. Now let $X$ be the $S$-scheme $S\times S$ via the first projection, $\text{pr}_1$, and let the section $\chi$ be the diagonal morphism. Next, let $(\rho:Y\to S, \upsilon:S\to Y)$ be the rank $1$ vector bundle over $S$ with $\upsilon^*\mathcal{O}_Y(\upsilon(S))$ isomorphic to $\mathcal{O}_S(2)$, and where $\upsilon$ is the zero section. These $S$-schemes with section are formally isomorphic as above. However, every nonempty, effective Cartier divisor in $S\times S$ intersects the diagonal $\chi(S)$. Thus, as in the above example, there is no $S$-scheme with section $(\sigma:U\to S,\tau:S\to U)$ and pair of étale $S$-morphisms, $f_X:U\to X$, $f_Y:U\to Y$ that is compatible with $\chi$, $\upsilon$, and $\tau$.

Second Edit. It has been pointed out to me that, in the global case, an infinitesimal extension that admits a retraction is not necessarily "split". Thus, it is unclear for the pairs above, whether or not the formal completions of $X$ and $Y$ along the image of $S$ are isomorphic.

ALERT. I was alerted to a problem with the following by an e-mail. Unlike the affine case, in the global case, the formal completion of a smooth $S$-scheme along a section may fail to split. Thus, in the examples of $X$ and $Y$ below, the formal completions may be non-isomorphic. In fact, likely they are non-isomorphic. So the following example is probably wrong.

There cannot be a "global" result in complete generality, but there certainly can be results that are "local" over $S$. Here is one global counterexample. Let $S$ be $\mathbb{P}^1$. Let $Z$ be $S\times S$, and let $\Delta:S \to Z$ be the diagonal morphism. Let $(\pi:X\to Z,\chi:Z\to X)$ be the vector bundle of rank $1$ such that $\chi^*\mathcal{O}_X(\chi(Z))$ is isomorphic to $\text{pr}_1^*\mathcal{O}_S(0)\otimes \text{pr}_2^*\mathcal{O}_S(2)$ as an invertible sheaf on $Z=S\times S$. Similarly, let $(\rho:Y\to Z,\upsilon:Z\to Y)$ be the vector bundle of rank $1$ such that $\upsilon^*\mathcal{O}_Y(\upsilon(Z))$ is isomorphic to $\text{pr}_1^*\mathcal{O}_S(1)\otimes \text{pr}_2^*\mathcal{O}_S(1)$.

Edit. I realized that there is an easier example underlying the example above. Again let $S$ be $\mathbb{P}^1$. Now let $X$ be the $S$-scheme $S\times S$ via the first projection, $\text{pr}_1$, and let the section $\chi$ be the diagonal morphism. Next, let $(\rho:Y\to S, \upsilon:S\to Y)$ be the rank $1$ vector bundle over $S$ with $\upsilon^*\mathcal{O}_Y(\upsilon(S))$ isomorphic to $\mathcal{O}_S(2)$, and where $\upsilon$ is the zero section. These $S$-schemes with section are formally isomorphic as above. However, every nonempty, effective Cartier divisor in $S\times S$ intersects the diagonal $\chi(S)$. Thus, as in the above example, there is no $S$-scheme with section $(\sigma:U\to S,\tau:S\to U)$ and pair of étale $S$-morphisms, $f_X:U\to X$, $f_Y:U\to Y$ that is compatible with $\chi$, $\upsilon$, and $\tau$.

Second Edit. It has been pointed out to me that, in the global case, an infinitesimal extension that admits a retraction is not necessarily "split". Thus, it is unclear for the pairs above, whether or not the formal completions of $X$ and $Y$ along the image of $S$ are isomorphic.

Edit. I realized that there is an easier example underlying the example above. Again let $S$ be $\mathbb{P}^1$. Now let $X$ be the $S$-scheme $S\times S$ via the first projection, $\text{pr}_1$, and let the section $\chi$ be the diagonal morphism. Next, let $(\rho:Y\to S, \upsilon:S\to Y)$ be the rank $1$ vector bundle over $S$ with $\upsilon^*\mathcal{O}_Y(\upsilon(S))$ isomorphic to $\mathcal{O}_S(2)$, and where $\upsilon$ is the zero section. These $S$-schemes with section are formally isomorphic as above. However, every nonempty, effective Cartier divisor in $S\times S$ intersects the diagonal $\chi(S)$. Thus, as in the above example, there is no $S$-scheme with section $(\sigma:U\to S,\tau:S\to U)$ and pair of étale $S$-morphisms, $f_X:U\to X$, $f_Y:U\to Y$ that is compatible with $\chi$, $\upsilon$, and $\tau$.