Once we correct the definition of $[×2]$ to make it a functor (currently it does not preserve identities) by setting $f'=f⊔f$, it is true that the construction described in the main post gives rise to an equivalent notion of homotopy, assuming $Y$ is Kan.
Indeed, $[⨯2]$ is a right Quillen functor from simplicial sets to simplicial sets, with its left adjoint being the unique left Quillen functor that sends $Δ^n$ to $Δ^n\ast Δ^n$, the join of $Δ^n$ with itself. We also have canonical inclusions given by the adjoints of projection maps in the main post: $X→X\ast X$ can be included as the left or right factor. The induced natural transformation $X⊔X→X\ast X$ exhibits $X\ast X$ as a (generalized) cylinder object for $X$, where instead of a genuine codiagonal map $X\ast X→X$ we only have a zigzag $X\ast X→Y←X$, where $X→Y$ is a weak equivalence. But this is sufficient to answer the original question in the affirmative.
Equivalently, $Y[⨯2]$ together with the two natural maps $Y[⨯2]→Y$ is a (generalized) path object. A bit of elementary model category theory (see the cited articles) then implies that the notion of homotopy induced by this new cylinder (or path) object coincides with the old notion, assuming $Y$ is Kan.