How can an electron being a wave have such property as spin?
An electron is not a wave nor a particle. An electron is usually described as a quantum object with some wave-like properties and some particle-like properties.
Some of them, like particularly a spin, do not have direct counterpart in our familiar macro world. It is a kind of a mysterious, specifically quantum property of all elementary particles and  atomic kernels, that interact with angular momentum and interchange with it its values.
Non-relativistic quantum mechanics has no theoretical idea/model about the observed spin phenomena, that would come e.g. from resolving the Schrödinger wave equation. It was  added just as an ad hoc extension. OTOH, the relativistic Dirac wave equation brings spin as the natural part of the quantum mechanical model.
Note that the classical idea about a spin as a rotation related angular momentum of a fast spinning small ball is wrong. Considering the electron mass, energy and the limit of the maximum electron size, an electron simply cannot rotate fast enough to have such amount of classical angular momentum, related to its eventual spinning.
Both orbital and spin contributions to the total angular momentum have their counterparts as orbital and spin magnetic momentum, if quantum objects have nonzero electric charge or charge nonhomogenity (like neutrons ).