Unique/exotic Japanese food?

Yes, Japan has tons of "weird" dishes from a Western POV. Visitors to our Tokyo office are regularly fed one or more of raw chicken (torisashi), whale (kujira) and fish milt (shirako), and these are at the milder end of the spectrum. For really weird stuff (read: inedible to the average Westerner), check out Japan's wide range of wacky seafood products: ika no shiokara fermented squid pickle, konowata sea cucumber guts, hoya sea pineapple ("like rubber dipped in ammonia"), etc. Here's a nice checklist of some of what I've run into over the years: https://driftingclouds.net/2007/01/02/weird-food-compendium/

As for where to find them, your best bet is to find an izakaya (Japanese pub) known for its chinmi ("rare tastes"), which will have a revolving gallery of whatever is good (and unusual) in this season.

Japan certainly has some unique foods. One famous/notorious example, which even the most adventurous eater might think twice about, is Ikizukuri:

ikezukuri (????), (roughly translated as "prepared alive") is the preparing of sashimi (raw fish) from live seafood. In this Japanese culinary technique, the most popular sea animal used is fish, but octopus, shrimp, and lobster may also be used.

The practice is controversial owing to concerns about the animal's suffering, as it is still alive when served.

...The restaurant may have one or several tanks of live sea animals for a customer to choose from... Ikizukuri fish may be prepared with only three knife cuts by the chef. They are usually presented with the head still whole so that customers are able to see the continuing gill movements.

Similar traditions exist in China and Korea (film fans might remember the live octopus eating scene from the original Oldboy) but there are some major differences described in this blog:

• the Chinese have their own way of Ikizukuri, deep-frying the fish and keeping them alive till they are eaten, known as Yin Yang fish

• In Korea, this practice is less cruel. Sannakji which are small octopuses cut into little pieces and usually lightly seasoned with sesame oil and sesame. While they are already dead, their nerves and suckers are still active, causing them to squirm and stick to the insides of your mouth. I’ve had the opportunity to try this, and let’s just say there wouldn’t be a second time. One [has] to be careful to chew them carefully as several cases of choking have been reported as the suckers stick to your throat when swallowing

Curiously, the foodstuff of Japan that presents the biggest danger and causes (I believe) the most deaths is actually one of the most seemingly ordinary: extremely sticky mochi rice cakes, which are easy to choke on and difficult to dislodge, and cause several deaths every year, usually around New Year celebrations.

A few of the more "prevalent" exotic foods of Japan:

• Horse meat, which the Japanese dub sakura-niku "cherry blossom meat" for its pinkish color. It is best eaten raw as basashi - making it even more exotic. Usually considered a delicacy of Kumamoto Prefecture, basashi is now more or less available around the country.
• Salted and fermented seafood, prevalent in the Hokkaido and Tohoku (less sure about this) regions of the country. One example would be the shiokara, salted & fermented squid innards.
• Live squid/octopus/shrimp(?) sashimi.
• Natto, or fermented soybeans. They smell, um...pretty bad.
• There is a wide selection of chicken/pig/ox innards available at yakitori (Japanese kabob skewer) restaurants - which range from liver (most "normal" IMO) to pig uterus (which I would rather not try).
• I know that insects are delicacies in some prefectures of Kyushu and Shikoku, but I am not entirely sure about their consumption. If you visit those regions, you might want to give a try.
• Pig ears are eaten in Okinawa.