Great question, Iakov. The issue is what properties a "book" can have, as apposed to what properties "books" can have. A (single) book can have any of these properties: kind, style, length, author, color, weight, price, etc. On the other hand, (many) "books" can have a wider variety of properties because they are many. For example, books can have "colors", or also "color". It would be strange and unlikely for many books to have the same color, but it is not impossible, so you can talk about the "colors" (if that is what you intend) or the "color" of books. The same considerations apply to "kind". "Kinds of books" seems more appropriate in most situations but it is not wrong to say "kind of books" if that is your intention.

The singular and plural forms of "beetle" are the same when referring to types, and you can also write it as beetles.

Subjectively, "type" implies a more objective and convincing way of categorization. When you are talking about something, it is usually on the basis of a certain data standard as a distinction.