Why do nouns in German have a gender?
The simple answer is that in German, all nouns are assigned a gender, either masculine, feminine or neuter. There are some exceptions to this rule (e.g. certain foreign words), but for the most part, all German nouns have a gender.
There is no real reason why the genders are assigned as they are; it’s just something that has developed over time. In some cases, the gender of a noun can be determined by its meaning (e.g. masculine: “der Mann” [the man], feminine: “die Frau” [the woman]), but often there is no logical explanation for the gender of a particular word.
One theory is that the genders of German nouns were originally based on the grammatical function of the word in a sentence (e.g. subject vs. object). For example, the word “der Tisch” (the table) is masculine even though it typically plays the role of an inanimate object in a sentence. On the other hand, the word “die Frau” (the woman) is feminine even though it typically plays the role of an animate subject in a sentence.
Another theory is that the genders of German nouns were originally based on the biological sex of the reference of the nouns. For example, the word “der Hund” (the dog) is masculine even though the vast majority of dogs are male, and the word “das Mädchen” (the girl) is neuter even though the vast majority of girls are female.
A third theory is that the genders of German nouns were originally based on the animacy of the referents of the nouns. For example, the word “der Baum” (the tree) is masculine even though trees are inanimate, and the word “die Katze” (the cat) is feminine even though cats are animate.
It is difficult to say definitively which of these theories is correct, or if all of them are partially correct. What is certain is that the genders of German nouns are not always predictable, and that native speakers simply have to learn the gender of each noun individually.