"Isotopic space" means the internal space you move around in when you do Isospin transformation. For a Nucleon, "Isotopic space" consists of all superpositions of proton and neutron, for a pion, it's all superpositions of pi0 pi+ and pi- and so on. Because of the history of gauge theory, Yang and Mills were gauging the Isospin transformation, any gauge group is sometimes called isotopic space even if it has nothing to do with isospin.
The Pauli matrices are SU(2) generators, SU(2) and SO(3) are the same Lie algebra. The 'tHooft Polyakov monopole, you have an isospin vector (it's a collection of 3 scalars which form a vector under isospin rotations) which is "radial" (or a "hedgehog" in Polyakov's terminology), meaning that
$V_1 = x f(r)$
$V_2 = y f(r)$
$V_3 = z f(r)$
where $V_i$ are the three vector isospin components (they rotate into each other when you do an Isospin rotation, nothing happens to these components when you do a spatial rotation), and x,y,z are space indices (they rotate when you do an actual rotation, nothing happens to space when you do an isospin rotation).
When you do a rotation of space on a configuration of scalar fields like this, the field is not invariant, but you can fix that if you do the same isospin rotation also, independently, at the same time. That means the solution breaks two independent SO(3)s to one SO(3).