It doesn't. In the screenshots I posted, Swampert has both Physical (Earthquake and Rock Slide) and Special attacks (Ice Beam and Hydro Pump). The first one shows the base damage both Physical and Special would do without a choice band, and the second one shows the damage of both Physical and Special types with a Choice Band. If you compare the two, only the Physical attacks received a boost.
To answer the question directly again: It isn't both. It only raises physical attack, not special attack.