Why three families of multipole moments? There are three families of multipole moments: The electric multipole moments, the magnetic multipole moments and the toroidal multipole moments. Is there any reason why there are this three families and not two or four or even more? Are there for example symmetry reasons or something like this?
 A: Your guess is correct.  That classification of order $n$ multipole moments can be viewed as a symmetry classification of rank $n$ tensors, so how many of them there are is a matter of what symmetries exist in spacetime.
There are two types of "inversions" that can be performed on spacetime's two types of physical dimensions: a spatial inversion, also called a parity inversion, and time reversal.
The lowest-order multipole moment that's still interesting is the dipole moment, which is represented by a vector (a rank 1 tensor).  And the symmetry properties of a dipole moment can be classified by the four possible symmetry characteristics of a vector under the two types of spacetime inversions.
The following table, which was lifted from Wikipedia, shows the symmetry properties under inversions of the various kinds of dipole moment.  In this table, a -1 in the P column means that the dipole moment changes sign under a parity inversion, and a +1 in the P column means that it doesn't change sign under a parity inversion.  Similarly, a -1 or +1 in the T column means that the dipole moment does or does not change sign under a time reversal.

The symmetry properties of different kinds of higher-order multipole moments similarly relate to the symmetry properties of higher rank tensors.  For a (rather advanced) discussion of the symmetry of some of the higher-order multipole moments, see this paper.
