It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two singular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

EDIT: By an elliptic fibration $\pi\colon X\to S$, I understand:

$X$ is a smooth algebraic surface. $S$ is a complete smooth algebraic curve over an algebraically closed field, $\eta$ is its generic point. $\pi$ is a projective morphism, and the generic fiber $X_\eta$ is a geometrically integral smooth algebraic curve of arithmetic genus $1$.

yes. Most cases consist of a collection of smooth genus $0$ curves meeting in some configuration of points. As there is a unique smooth genus $0$ curve over $k$ up to isomorphism (namely $\mathbb{P}^1$), the result is clear. The remaining cases are that of a plane cubic nodal or cuspidal curve. Again, such a curve is unique over $k$ up to isomorphism, hence the result follows. $\endgroup$ – Daniel Loughran Mar 16 '16 at 12:22