Magnets are macroscopic objects described by classical electricity and magnetism , Maxwell's equations.
Electrons are elementary particles posited in the standard model of particle physics axiomatically, i.e. their properties have been measured experimentally and are accepted as input to the theory. They are quantum mechanical entities.
All classical theories emerge from the underlying framework of quantum mechanics.
What I don’t understand is how can a fundamental particle( say electrons) spinning around its axis generate a magnetic field?
The classical language is used, but the particles in the standard model table are point particles, their spin and charge have been measured by painstaking experiments, using quantum mechanics and conservation laws that the QM solutions have to obey. Conservation of angular momentum is the method of deriving the spin of elementary particles. The word "spin" does not imply spinning as for a macroscopic object. It is there as an identifier of conservation of angular momentum at the quantum mechanical level. (It is the same as the orbitals of the electrons, not orbits, which are probability loci for the electrons in an energy level of an atom.) The resultant magnetic moment is not the result of orbits, but of the probability distribution of the orbitals of the electron.
And why are they known as tiny magnets( if so then can ‘North Pole and South Pole of this magnet’ be defined?
It is an analogy to the macroscopic magnets. Yes, north and south may be defined because they have two states of spin, +1/2 -1/2 . One has to keep in mind the quantum versus classical distinction though.
If special relativity is applicable, then how can it explain this phenomenon because it explains magnetic force as a result of length contraction between 2 moving wires.
Special relativity is paramount in the quantum framework, and the corresponding equations for generating magnetic fields exist, using the defined probability loci of the solutions for the equations.
Atoms and molecules and magnetic domains smaller than nanometers are modeled in the quantum mechanical frame. Long range behavior is built up by these underlying quantum states.