In Görtz&Wedhorn’s Algebraic Geometry I: Schemes, there is a definition:
Definition 3.46. Let $X$ be a scheme. For two subschemes $Z$ and $Z^{\prime}$ of $X$ we say that $Z^{\prime}$ majorizes $Z$ if the inclusion morphism $Z \rightarrow X$ factors through the inclusion morphism $Z^{\prime} \rightarrow X$
We sometimes write $Z \leq Z^{\prime}$ or simply $Z \subseteq Z^{\prime}$ if $Z^{\prime}$ majorizes $Z$. This defines a partial order on the set of subschemes of a scheme $X$.
But it seems that the antisymmetry property does not hold? From $Z \leq Z^{\prime}$ and $Z' \leq Z$ one only concludes that $Z \cong Z'$ but not $Z = Z'$, for even the underlying topological spaces are set-theoretically equal but the structure ring spaces can still be only isomorphic but not set-theoretically equal? It does be partial order if considering only as the skeletal category, but in this way it suffices to consider immersions but not subschemes?
The definition of subscheme Görtz&Wedhorn using is
Definition 3.43.
(1) Let $X$ be a scheme. A subscheme of $X$ is a scheme $\left(Y, \mathscr{O}_Y\right)$, such that $Y \subseteq X$ is a locally closed subset, and such that $Y$ is a closed subscheme of the open subscheme $U \subseteq X$, where $U$ is the largest open subset of $X$ which contains $Y$ and in which $Y$ is closed (i.e. $U$ is the complement of $\bar{Y} \backslash Y$ ). We then have a natural morphism of schemes $Y \rightarrow X$.
(2) An immersion $i: Y \rightarrow X$ is a morphism of schemes whose underlying continuous map is a homeomorphism of $Y$ onto a locally closed subset of $X$, and such that for all $y \in Y$ the ring homomorphism $i_y^{\sharp}: \mathscr{O}_{X, i(y)} \rightarrow \mathscr{O}_{Y, y}$ between the local rings is surjective. Whenever $Y$ is a subscheme of $X$, then the natural morphism $Y \hookrightarrow X$ is an immersion.
I found Why does the definition of an open subscheme / open immersion of schemes allow for an "extra" isomorphism? similar but not the same too.
