By asking "how many," it automatically makes the subject plural. It can't be "How many kinds of book do they stock?" because "book" is not and cannot be the plural for "books" in any context. You could say, "How many kinds of food do they stock?" because "food" is a word that by itself, can include several types of food. "Book" cannot be used the same way. "Book" can only be used as a singular noun or if you say, "They only have one kind of book."
its plural form only
It seems that British speakers accept either, while American English speakers go for the plural. Is this assessment about right?
Many kinds is in the plural form so we can't use book which is a singular.