I would definitely say the Yamato has some sort of place in the hearts of the Japanese people even today. In the late 30s, when it was laid down it was praised for being the biggest battleship in history and at the time many people still thought that size was the most important feature for these kind of Battleships. In theory it outgunned and outranged any US or British Battleship it would have faced, which in the thinking of WW1 or the Russo-Japanese War was the key to victory. History taught us that it wasn't enough, although the Yamato also had design flaws, as /u/kieslowskifan pointed out in this earlier thread.

While the Japanese were proponents of naval aviation themselves, they underestimated just how vulnerable giants like the Yamato would be to attacks from the air.

The whole concept of Battleships proved to be flawed in WW2, as can be seen in many other exemplary cases like the Bismarck, the Roma or the Sinking of the British ships 'Prince of Wales' and Repulse (by the Japanese themselves). And although many famous ships were essentially a waste of resources there is often still an almost mythological meaning to them in their respective countries, just for what they represented at the time and how tragic their loss was. ('Bismarck' for Germany, 'Hood' for Britain, 'Arizona' for the USA ...)

The Yamato and the Mushashi were huge battleships, dubbed unoffically "superbattleships" because of their 60k+ tonnes displacement and hugely powerful armament. The Ultranationalists of Japan, while not restricted by the League of Nations' naval treaties, greatly pushed for the expansion of the navy, including the creation of the Yamato-class battleship. Thanks to its size, power and speed, it's a no brainer that such gigantic and impressive looking ships came to represent the might of Imperial Japan, especially with the Empire's fierce defiance to the Western powers, obviously the US being the big one. It became a symbol of pride for all Japanese people

The Japanese knew that they couldn't go against the US naval output, so the Yamato was designed to be a hugely powerful battleship that could engage multiple targets. But as history has shown, battleships were in their twilight, this was the era of aircraft carriers. Both the Yamato and the Mushashi were sunk by aircraft carriers, and the Shinano, the third of its class, was in the middle of being converted into an aircraft carrier.

Even decades later, the symbol of Yamato remained strong. One of the most popular animes, Space Battleship Yamato, obviously being named after the ship. Thanks to the popularity of the series, the name Yamato was immortalized for another generation.

The short answer is yes, this particular battleship has been romanticized in postwar Japanese. The mystique of the Yamato is partly a function of its size, which is also a draw for non-Japanese audiences as well. However, the Yamato taps into a number of important postwar Japanese cultural archetypes that give the ship a unique staying power in public consciousness.

Although the ship itself had little military impact on the war, the Yamato's historical narrative meshes well with the larger historical narrative of Japan's defeat. The IJN intended the ship to be the expression of Japan's technical expertise and ability to transcend superior numbers to achieve victory. In this goal, the IJN clearly failed and Yamato paid the price. The Yamato became an analogue of sorts for the postwar memory and interpretation of Japanese defeat: Japan thought it had the means to win, but discovered its reach far exceeded its grasp. While other IJN ships had a more colorful service, few match this specific narrative. Films like 2005's Otoko-tachi no Yamato, based upon a bestselling nonfiction book, hew to this narrative, where its main protagonists, boy sailors, genuinely believe that the Yamato will lead Japan to victory and most of them die during the ship's sinking. In this way, the ship acts as a metaphor for Japan's whole wartime effort and erases some of the uncomfortable portions of its wartime aggression by recentering a narrative upon Japan's hubris (which can be hoisted on a few military leaders) and eventual victimhood.

In Ivan Morris's 1975 classic, The Nobility of Failure he argues that Japanese culture celebrates heroes who fail and yet stay true to their spirit and goals. While this is only one strain of Japanese culture, the story of the Yamato meshes well with this cultural archetype. In the aforementioned Otoko-tachi no Yamato, the ship's officers have an acrimonious meeting with the ship's commander over Operation Ten-Go. Many note, with great justification, that the ship's mission to Okinawa is a waste and the stuffed shirts at headquarters will not be called upon to make the same sacrifice. The ship's commander quiets down these objections by noting that the sacrifice of the ship will be a lesson for the Japan of the future, much like how late-Tokugawa samurai who encountered Western guns showed that the old samurai ways were not effective, so to would Yamato's sinking. The ship's crew bonds over the course of the film and most die in the end. There's a similiar narrative in the flashback portions of the anime Space Battleship Yamato where the crew stoically face defeat.

Space Battleship Yamato brings up another aspect of Yamato's postwar cultural transformation: the ability of fiction to transform defeat into a victory. In this particular anime, the wreckage of the Yamato become a starship that saves the earth multiple times. William Ashbaugh cogently argues that this particular anime refights the war in a way that Japan wins, albeit in this case Japanese stoicism and technical expertise saves humanity rather than advances imperialist goals. The animation of the final space battle mirrors the animation of the ship's 1945 one, although this time, the Yamato triumphs over superior odds. Similarly, the technothriller anime/manga Silent Service has a JMSDF nuclear submarine named Yamato go rogue and triumph over the more numerous American and other antagonists. In this way, fiction is reversing the trauma of defeat and crafting an alternative narrative wherein defeat led to a new victory.

Although the Yamato has a certain cachet in Japanese public consciousness, it is important to realize the caveat that reception is a different animal than transmission. The symbolism of turning defeat into victory of Space Battleship Yamato might be irrelevant to Japanese audiences who simply enjoyed a rollicking space opera. There is also a degree of apathy among the wider Japanese public about the Second World War and its relevance to contemporary Japan. Otoko-tachi no Yamato had a respectable box-office run but came in well behind Western films like Revenge of the Sith and Harry Potter and the Goblet of Fire. There's a clear Japanese market for Yamato merchandise, as evidenced by HobbyLink Japan's search result for "Yamato" or videogames like Kantai Collection. Yet what Yamato means for the larger Japanese population as a whole in 2015 is somewhat nebulous.

Maybe im a bit too late but I can try my best to give out a few answer.

As you said, the name Yamato or in kanji, 大和 carried a bit of cultural weight as it means Big Japan. As what the other users in the thread has pointed out, the super size of the battleship is the personification of the national pride. The Japanese navy played the important part of its propaganda.

Japan, being an island nation similar to Britain knows that having a strong navy means they are pretty much living in a fortress. Of course times changed and the nature of navy fighting has already evolved.

The modern culture of Japan still revered greatly at Yamato. Example is the anime, Space Battleship Yamato.