There is a photo on page 5 of this thread:
Birthday Tenkara grayling about 3/4 of the way down the page that shows the tip of a tenkara rod and the tip of a whip for comparison. It pretty clearly illustrates the difference in sensitivity.
A tenkara rod has a very thin, very flexible tip so that you can easily cast a very light line (.285mm fluoro works nicely) and also protect a light tippet. People catch 20-24" trout on 5x tippets.
Since your question was about "tenkara style" fishing, the answer is yes, but...
You can fish "tenkara style," but you will not be able to do it as successfully. Using a whip would almost certainly require using a heavier line to be able to cast well. A heavier line will be harder to hold off the water's surface, so drag will be much more of an issue. One of the biggest advantages of tenkara is that the light line can be held off the water - producing very little drag. Also, I suspect a whip will not be as forgiving when using light tippets.
The question (and answer) is about the same as asking if you could use a _____________________ rod for __________________ fishing (where the type of rod and type of fishing are not the same. The answer is yes you could, but why would you?