Whenever reading about formal groups, this axiom has always appeared to me as a bit artificial, at least compared to the other axioms.
To explain what I mean, suppose that $R$ is a ring, and that we are given a one-parameter (formal) group scheme structure on $R[[X]]$, the ring of formal power series in $X$. If we put $G = \text{Spf }(R[[X]])$, the multiplication $G\times G\to G$ corresponds to a continuous map of topological $R$-algebras (the comultiplication):
$$R[[T]] \to R[[X]] \widehat{\otimes} R[[Y]] = R[[X,Y]].$$
The image of $T$ in $R[[X,Y]]$ is a formal power series $F(X,Y) \in R[[X,Y]],$ and the axioms for a group scheme translate into the usual formal group axioms for $F$, that is, except the axiom $F(X,Y) \equiv X + Y \pmod {\langle X,Y\rangle^2}$. So, what is the reason for throwing it in? Is it just because the formal groups which appear "in nature" satisfy it?