The dark matter halo is not a solid object, but consists of many particles which move on various orbits in the gravitational potential of the Milky Way. Therefore, what is meant when talking about the speed of the sun with respect to the dark matter halo is the average relative speed between the dark matter particles in the vicinity of the sun and the sun itself.
Particles whose orbits pass through the vicinity of the sun certainly have a comparable orbital velocity (of course it depends on the orbital eccentricity), but from numerical simulations of galaxy formation it is thought that the tangential components of the velocities are approximately evenly distributed in all directions and so the average rotational velocity of the halo is quite low.
For more details you might look at e.g. Evans & Bowden (2014) which discusses among other aspects how far the Milky Way’s halo can deviate from the above spherical shape and still be consistent with observations. They place a lower limit of 0.6 on the ratio of polar axis to major axis.