There are multiple surviving traditions of European swordsmanship with continuous lineages going back many centuries (at least foil/epee/sabre fencing and military sabre). It has been, and still is, taught in schools. It is not at all true that European swordsmanship was forgotten and had to be rediscovered through fencing manuals. What is true is that European Medieval swordsmanship had to be rediscovered through fencing manuals.
As the common swords in use in Europe changed over time:
Military swords: arming sword -> war swords, longswords, two-handed swords, backswords, sabres, hangers -> late 19th and 20th century cavalry and infantry swords -> modern ceremonial military swords
Duelling swords: arming sword -> longsword -> rapier -> smallsword -> duelling epee and duelling sabre
the styles of swordsmanship changed. European swordsmanship was a living and evolving art, adapting itself to the swords currently in use. Once Medieval swords were no longer in use, the use of Medieval swords was no longer taught. But European swordsmanship changing to keep up with the swords in current use is far different from European swordsmanship being forgotten.
Since Japanese swords changed less over that time (the katana was in use from about 1400 through to the end of the Edo Period), there was less need to Japanese swordsmanship to evolve as much. Japanese swordsmanship wasn't static over this time - for an overview of its evolution, see G. Cameron Hurst III, Armed Martial Arts of Japan: Swordsmanship and Archery, Yale University Press, 1998.
Returning to Medieval European swordsmanship, one should note the survival as a continuous living tradition of the Portuguese stick-fighting martial art, Jogo do pau. The origins of Jogo do pau are lost to history, but based on technique and terminology, Medieval swordsmanship is one of its ancestors.
The majority of historical Japanese martial arts, koryu bujutsu, that exist today were not directly applicable to the large-scale battles that occurred during Japan's medieval period (Warring States period). Furthermore, the organizations that have transmitted these martial teachings, called ryuha or ryugi, are mostly a product of Japan's early modern period (Azuchi-Momoyama through Tokugawa periods). Because these ryuha were formed anachronistically for reasons other than direct application in pitched battle, their perceived value was never tied directly to the practicality of the technique and weapons they taught. For more information see Karl Friday, Off the Warpath: Military Science & Budo in the Evolution Ryuha Bugei, Budo Perspectives, 2005.
Because martial arts provided value beyond just fighting, they and the ryuha that transmitted them were widely adopted as standard education for the warrior class throughout the early modern period. In some cases, martial arts training extended beyond vocational training of the warrior class and was adopted by non-warrior classes as sport, hobby, and community involvement. This was particularly true of grappling and small arms martial arts. For more on this latter development, see Shoji Enomoto, A Historical Study on the Muso Jikiden Schools in North Shinano, Budogaku Kenkyu, 1994.
However martial arts training and education was dealt a number of damaging blows during and after the Meiji Restoration. As regional governments were restructured via the Abolition of the Han System, haihan-chiken, in 1871 many traditional schools lost patronage. Swordsmanship in particular further suffered from the Sword Abolition Edict, haitorei, of 1876. However, because difficulties adopting western physical and martial education and coinciding the with enacting a Conscription Act, choheirei, in 1883 the Ministry of Education, monbusho, issued an order to the taiso denshu-jo (Place of Physical Education) to conduct a survey on the educational merits of Japanese swordsmanship and grappling. As a result, traditional Japanese physical education, in the form of martial arts, returned to the education system's official curriculum. These changes were further spurred on by the first Sino-Japanese War and Russo-Japanese War and led to a widespread return of martial arts education, both modern and traditional, that continued until the banning of martial arts practice by the GHQ after WW2. See Sugie et al, An Inquiry on the State of Kendo Transmission in the Tokai Region during the Meiji Period, Budogaku Kenkyu, 1971 and Nobuhiro Morita, The Encouragement of Budos in Akita Prefecture in Meiji Era, Akita Univesity, 2003.
In summary, extant historical Japanese martial arts, including swordsmanship, have always had a place in Japanese society beyond simply teaching men to fight. However, the practice of these martial arts has waxed and waned and has undoubtedly changed in the process. Although the means and results of such change may be different in detail from its European counterpart (described by /u/wotan_weevil), the overall story is fairly similar.
It's important to point out that kendo is very similar to fencing in that both of these are mostly sports rather than a concentrated effort to recreate a life or death duel. So both kendo and fencing are in the same realm in that their techniques, while having a lot of crossover to their heritage, are not really focused on martial combat.
What is being done with WMA/HEMA is to recreate techniques in their proper martial and historical context rather than being limited to the ones useful to a sport. And of course, a lot of those techniques are limited to a specific subset of swords. IE, you can't do the same things with a rapier as you could with a longsword due to the handguard.