In dimensions at least 4, one has that homotopy of circles implies isotopy of circles. So if you fix an embedding of $S^2$ (eg the standard embedding), then you are really asking about embeddings of the circle in the complement of that fixed $2$-sphere, ie about the fundamental group of the complement. If you fix the standard embedding, then the fundamental group is $\mathbb{Z}$. (By the way, these groups are typically not abelian, so you have to fix a base point in order to get a group.) The same comments would apply if $4$ is replaced by $n$ and $n-2$ by $2$.

I don't really understand what you mean more generally about the group structure on $Emb(S^1 \sqcup S^2,S^4)$. Beyond the need for a base point to use in multiplying circles by connected sum, there doesn't seem to be an inverse with respect to connected sum of $2$-spheres.