Why is the alpha particle called a particle when it is made of four particles?
Actually, the alpha particle contains two protons and two neutrons. It is what is emitted from the nucleus under alpha decay, giving off what was first classified as alpha radiation by Rutherford at the end of the 19th century (very late 1800's).

In 1932 the neutron was discovered. Only then did it become clear that the nuclei of atoms contain more than just some positively charged particles (which we know as protons).

So there is this historic reason for the naming of the alpha particle.

"Particle"

The word particle does not only pertain to elementary particles (and in [quantum] chemistry, we stop at the level of nuclei as particles, everything that goes deeper is nuclear physics). There are smoke particles, dust particles, etc. 

There is also the wave-particle duality. In principle, since we can observe this "alpha radiation" from radioactive processes, there must be a particle that belongs to the wave. With electromagnetic radiation, this is the photon, and in alpha radiation it is the alpha particle.