First, you are asking about Ampere's law in magnetostatics, and is definitely a valid question.
Second, you're right. Both sides of this equation are volume-based quantities. But when converting to integral form by using Stokes' theorem, you have to pick a contour and the surface enclosed by the contour. If you pick the contour as a circle rotating around the wire, then you arrive at the integral form the Ampere's law. The volume current density though may point in any direction, but only the portion that goes through the surface ends up contribute.
You are right, the curl of the magnetic field seems like zero everywhere for a wire current (when you directly calculate the differential value), but it is not zero at the wire location. You can apply Stokes theorem in cylindrical coordinate for a small circle around the wire to find it out. It is equal to I/S, where S is the area. When you make S infinitely small, it is approaching infinity. Therefore, the curl for the H field of an infinite wire at the wire location is a delta function along the wire. It is zero everywhere, except for when collocated with the wire.