Aina is indeed a name, although I don't know how "Swedish" it is. Two famous Ainas are Aina Erlander (Wife of Swedish Prime Minister Tage Erlander) and Aina Berg (Famous swimmer), but the name is popular outside of Sweden as well.

But I do agree that probably today "Aina" is more familiar to people (born after the 90s at least) as a nickname for the police. First times I came across it I think was from Swedish gangsta rap songs, where the phrase "Aina, La hora" seems to be a popular phrase, something that I think would be translated to "The cops, The whores" or something like that.