Adding to the previous answer: Actually in its original form due to Eichler and Kneser one uses not $Pin$ but the spin group, which is the two fold simply connected cover of the special orthogonal group and has its name from its use in quantum theory.

The term genus is an english version of the original "Geschlecht" introduced by Gauss. As already remarked, this is just a notion of grouping things together.

Adding to the previous answer: Actually in its original form due to Eichler and Kneser one uses not $Pin$ but the spin group, which is the two fold simply connected cover of the special orthogonal group and has its name from its use in quantum theory.

The term genus was introduced by Gauss (genus in latin, Geschlecht in german). As already remarked, this is just a notion of grouping things together.