The definition of subobject in Wikipedia is this:
let $A$ be an object of some category. Given two monomorphisms
$u : S\to A$ and $v : T\to A$
with codomain $A$, we write $u\le v$ if $u$ factors through $v$—that is, if there exists $\varphi : S\to T$ such that $u = v \circ \varphi$. The binary relation $\equiv$ defined by
$u\equiv v$ if and only if $u\le v$ and $v\le u'$
is an equivalence relation on the monomorphisms with codomain $A$, and the corresponding equivalence classes of these monomorphisms are the subobjects of $A$. (Equivalently, one can define the equivalence relation by $u\equiv v$ if and only if there exists an isomorphism $\varphi:S\to T$ with $u = v \circ \varphi$.)
This article doesn't assign a name to the equivalence relation $\equiv$. Can we call it isomorphism, and say that $u$ and $v$ are isomorphic if $u\equiv v$, or is this confusing, because isomorphism is here the morphism $\varphi$, and the adjective isomorphic is reserved for objects, not to morphisms? Or does this $\equiv$ relation have some another name?