It is true that the tokamak exhibits an axisymmetric magnetic field and is basically 2D in nature, while the stellarator has a fully 3D shape (and is thus not axisymmetric). This difference is due to the fact how the confining magnetic field is created.
In both cases, you need twisted magnetic field lines to achieve confinement. The tokamak achieves this via a combination of a strong toroidal magnetic field, generated by field coils, and a poloidal magnetic field, generated by a strong current flowing in the plasma (plus a vertical magnetic field, generated also by coils). The resulting magnetic field is axisymmetric, as you have said, and thus basically 2D in nature. The drawback of this configuration is the required strong plasma current which generates the poloidal magnetic field.
The stellarator on the other hand generates the magnetic field only via magnetic field coils, such that there is no necessity for a strong plasma current. This requires, however, field coils which wind around the torus helically (to achieve the twist in the field lines). On modular stellarators this becomes more complicated as you use modular coils to mimic the helical winding. The increased number of coils (which can also be quite close to the plasma) generate a fully 3D magnetic field which has a strong modulation of $|B|$ in basically every direction.
The strong modulation of $|B|$ in a classical stellarator has a few consequences, mainly that transport losses are strongly increased (thus the confinement is decreased). This is what we call neoclassical transport (in contrast to classical transport which is due to collisions only).
Now we come back to your question (sorry for that lengthy introduction, but I have no idea about your background). Boozer  has developed the idea that those neoclassical transport losses depend on the variation of |B| within a flux surface, not on the vector components of B. These coordinates are nowadays referred to as Boozer coordinates. Nührenberg made use of this idea throughout the following years and proposed stellarator configurations that have a direction of quasi symmetry of $|B|$ in Boozer coordinates. Note that those configurations are still 3D in Cartesian coordinates (or more general in Euclidean space).
To answer your question: axisymmetry refers often to Boozer coordinates (or other, similar, magnetic coordinates); in Cartesian coordinates the shape of the flux surfaces might still appear three dimensional (and thus not axisymmetric).