What are the axes in the structure of an atom?

The shape of the orbitals are an artifact of the way Schrodinger's equation is solved. The first part of the standard solution is to pick a direction for the x,y,z axis and convert to polar coordinates.

The solutions we get reflect the chosen coordinate system. The distribution is actually isotropic if there is no preferred direction in space (no electric or magnetic field). The electrons are indistinguishable and in states that are "spherical" mixtures of orbitals (until a measurement is made/interaction happens). The picture is complicated by Pauli exclusion principle which constrains which mixtures are allowed at the same time. Statements such as "one electron is in the p_x orbital" are somewhat misleading and simplified, but much easier on high school Chemistry students.

Here is a semi-handwaving explanation.

Whenever we attempt to confine an electron in a small volume of space, we discover that the chances of finding it at any particular location within that space become nonuniform: there are some places where we are more likely to observe it than in other locations, and also some energy levels of the electron will become more likely than others. Furthermore, the smaller the space in which we try to confine it, the stronger these effects become. When we reach the size scale of an atom, the positional probability of the electron is highly nonuniform, and its energy levels have become discrete- which is to say that the electron can only hang out in certain regions, and we call those regions orbitals, and that hanging out somewhere inbetween two allowed orbitals is forbidden.

The electron itself resists being stacked into an orbital that already is occupied by another electron- unless we invert the spin of one of the electrons, in which case we can fit more than one of them into one orbital.

The lowest-energy orbitals look like spherical clouds, but as we try to put more and more electrons into position around a larger and larger nucleus, we discover that the electron's angular momentum is also quantized, and there are certain angular positions in space around the nucleus where those additional electrons are more likely to be observed than in others. These orbitals are not spherically symmetric- they stick out along certain preferred directions, and all higher-energy orbitals have to accommodate by avoidance all of those other orbitals.

So the facts that any given electron wants to occupy a state of minimum energy in the vicinity of a nucleus, that energy states or levels for electrons being confined by electrostatic attraction to the nucleus of an atom are discrete, that electrons cannot be stacked on top of each other unless their spins are opposite, and that their angular momentum is quantized, all give rise to the orbital structure as described in your course materials.

The positions of all those orbitals are relative to all the other orbitals- that is, there isn't anything in the nucleus itself that preferentially steers a given electron pair into two lobe-shaped volumes that stick out in opposite directions from one another; the nucleus appears to the electrons around it to be electrostatically spherical even though the locations of the allowed orbitals in space around the nucleus are not.