If I describe a $2$-torus in $4D$, as the product of two independent circles $S^1\times S^1$. Can the resulting torus live on the $4D$-embedded sphere $S^3$?
I want to confirm points of my torus can all be placed at a constant distance from the origin.
It can. There is a well-known embedding of $S^1 \times S^1$ in $\mathbb R^3$. See for example here. But $\mathbb R^3$ embeds into $S^3$. A standard embedding is given by the inverse of the stereographic projection form the north pole.