It does include dairy.
To help make your shopping in Japan simpler and a bit more certain, you should learn to recognize the word "allergy" in Japanese: アレルギー and the kanji for milk: 乳
I wouldn't say that all food manufacturers are including a list of contained allergens on all packages of all products, but recently a lot of packages, restaurant menus, and signage in places such as bakeries do have the information posted.
The information is often included in a little box of its own and should be pretty easy to spot. If you look carefully you should be able to find it on the package you are asking about here.