Now, aside from the obvious effectiveness of the sword as a combat weapon, the cultural significance of the sword in anime/manga/Japanese-produced video-game - whether a traditional Japanese sword or otherwise - is often cultural; the sword is a representation of the soul of the user - it is his/her very fighting spirit. As such, within the realm of Bushido, dating as far back as 712 A.D., under the rule of Emperor Mommu, and ending in 1880 A.D., during the Meiji Restoration period, Japan saw the installed rule of the samurai class, which were sworn to uphold both very explicit legal and moral codes regarding sword etiquette, such as calling for a samurai to be armed at all times, even in times of peace or formal events, or requiring samurai to clean and maintain their swords on a daily basis, and get this: due to the moral code of samurai regarding their preventing damage to their sheaths, or saya (鞘), of which death of the offender was deemed an acceptable punishment, samurai, wearing their swords on the left so as to be drawn with their right hand, would ritually walk on the left side of streets to avoid damage; during the modernization - and undoubtedly the Westernization - of Japan, when the automobile made its international debut, and paved roads were to be defined as either left-side or right-side, Japan, drawing from that particular samurai tradition, assented to a left-side road schematic. Moreover, the sword was the primary weapon of the Japanese warrior in combat, up until 1872 A.D., wherein they became a secondary weapon, as the gunpowder-based rifle began making its way into Japanese ports. Accordingly, whenever an anime/manga/Japanese-produced video-game features a traditional-warrior role, it is more often than not a sword wielder of some degree. All things considered, it is no surprise as to how or why the sword's importance has integrated itself into Japanese culture.
And yes, many swords appearing in these forms of Japanese-influenced media closely resemble these conventional styles - the Kill Bill movies immediately come to mind - but also do anime/manga series Samurai Champloo, Jubei-Chan, Blood +, Rurouni Kenshin, One Piece, Ninja Scroll, and Afro Samurai to name a few, as well as the Way of the Samurai, Bushido Blade, Battle Arena Toshinden, Samurai Shodown, Ninja Gaiden, the Soul Calibur and the pre-VII Final Fantasy video-game series, for the most part, all feature swords styles steeped in tradition. But as the world has come to find out, as much as Japan is known for upholding its traditions, the nation is equally analogous to, well, let's say innovation. But what seem to be more popular nowadays in Japanese and Japanese-influenced media than orthodox sword styles are examples of more contemporary sword styles ranging from marginally updated to downright outlandish can be viewed in a wide array of media: anime/manga series Bleach, Naruto, Storm Riders, Inuyasha, Samurai 7, Berserk, and a great many Gundam series, and any amount of video-games, like Star Gladiator, Guilty Gear, BlazBlue, No More Heroes, Strider, Kingdom Hearts, surprisingly Halo and the Metal Gear Solid series, the more recent Final Fantasy titles, with the exceptions of IX and XI, and the list goes on.
But why are swords so popular, as opposed to other weapons? Simply put, because they look cool, they're varied, they're very customizable, and they can cut through stuff - and really fast, at that. Additionally, due to the lore attributed to many of these blades, often owned and wielded by famous warriors, outlaws, military commanders, or generals, they are sought after as treasures - either personal or national - either because of the swords' exclusive rarity and/or extremely high-quality forging value. What also adds to their high demand are the legends that allegedly grant the wielder special abilities or powers, invisibility, life-stealing, or outright superiority over other blades - the stuff legends are made of, in other words.

If one could propose another weapon that fulfills so much of a history, warrior code, versatility in battle, variability in style and design, cultural impact, aesthetic significance, may it be mightier than the gun, but not as mighty as the pen.
-P. Strudwick