In SM, people say the ##SU(2)\otimes U(1)## theory unify the EM and weak interaction. I understand that the two interactions both originate from the electroweak theory in a mixed way. But does this mean they are unified? Since when we talk about unification like GUT, we require the couplings run to the same value at a high unification scale. I don't see it in electroweak theory. So how should we interpret the difference?
Yes, you can justifiably complain that the electroweak theory does not really unify anything since it still has two independent gauge groups, SU(2) and U(1), with independent couplings. Ideally we might have hoped that both forces would arise from a single gauge group. GUTs are more ambitious and seek to derive the SU(3), SU(2), and U(1) of the Standard Model from a single gauge group.
The weak and EM couplings are the same alpha=1/137. However the weak interaction depends on alpha/(M_W^2+Q^2), while EM is alpha/Q^2, so the weak looks weak at low energies.