"Are Chinese dumplings and Japanese dumplings the same?"
If you use 是, then you should use a real adjective after it, like this:
Yes. You can see from the answer given above that "一样" is used like a verb. That is a very interesting feature of Chinese. In Chinese, verbs and "adjectives" are very similar. And if you want an "adjective" to behave like an adjective, you need the marker "的".