Hey, very interesting question! To start, based off you question and how you are framing it, when you are referring to Japanese "castles", I believe what you are referring to is a tenshu, such as this image of the tenshu at Osaka castle in present day. There are 2 things to point out with regards to this. First, is that many of these (such as the one in the image above) are more modern reconstructions. These reconstructions often include building elements, and construction materials, that were not present in the period your talking about. For example, the tenshu at Osaka Castle uses concrete as a building material, and has an elevator. The occurrence of modern-day reconstructions of tenshu are the norm. The second point to bring up, and the one which will get at the heart of what you are asking, is that the tenshus were part of a comprehensive (and quite impressive) castle complex. This image shows a more zoomed out view of present day Osaka Castle shows part of the castle complex. Again, I should reiterate that the picture is not reflective of what you would exactly see in the period you are talking about, but gives a picture where we can form the basis of the exploration to why a tenshu would use wood as a primary building element.

Before moving forward, just some clarification in terms of what time period we will be discussing. The Japanese castles that you are referring to came to prevalence during the Sengoku period (15th C - early 17th C). This was a period which was defined by persistent warfare, and pervasive socio-political unrest. It is within these conditions that these sort of castles came into being. David Spafford in his monograph A Sense of Place: The Political Landscape in Late Medieval Japan describes this as somewhat of a "castle-building arms race" (pg. 170) that occurred in the latter half of the 1400s. As the period went on, castles grew more numerous, larger, and more complex (notable ones being Oda Nobunaga's Azuchi Castle, and the Hōjō's Odawara Castle). So what set these sort of castles apart from what came before it? Fortifications had been around previous to this, but they had a different function and built form. These were often less permanent fortifications that were built on mountains, and used situationally for defense. A critical point here is that a lord would not reside in the fortification. The lord would have a residence where they would rule from, and if such a need would arise, they would move locations to some sort of fortification for defense. What (partly) set aside the castles we start seeing in the Sengoku period is the joining of the functions of the fortifications and lords residence, so the castles became a place a lord would reside and rule from, as well as the location they would reside in when attacked. David Spafford in the aforementioned book describes this when stating:

"The castles built after about 1450, which I have been calling new castles, consolidated behind stronger, permanent ramparts the functions once divided between mansions and mountaintop fastnesses. These new castles were not simply further downhill than their earlier namesakes, they were fundamentally different, for by providing long-term defensive capabilities to residential and administrative complexes, they now became places of daily life and administration as well as refuge, embodying a warriors claims to both military and political power." - page 180

This starts to move us towards the question of why is the tenshu not made out of stone or masonry like we might see in European castles. The answer to this is twofold. One is that the defensive capabilities of Japanese castles of the period were built open the 'castle complex' itself, rather than the tenshu (a point I will elaborate on shortly). The second reason is that the tenshu form was driven (in part) from it purpose. And this purpose was to act as the physical representation, and administrative center, of a daimyo's holdings. Morgan Pitelka has a fantastic book titled Spectacular Accumulation: Material Culture, Tokugawa Ieyasu, and Samurai Sociability where he discusses how varying acts and occurrences (such as the mustering of large armies, acquisition of famous tea ceremony items, gift giving, head hunting, and falconry) were ways in which the social structure of samurai were formed, molded and represented. The construction of inspiring tenshus served this purpose. They were a means in which a daimyo or lord could articulate their power, in a built form for observers to witness. Oleg Benesch and Ran Zwingenberg highlight this condition in their book Japan's Castles: Citadels of Modernity in War when stating:

Unlike walls, ramparts, bridges, and other castle components, tenshu fulfilled little or no practical military function throughout their history. Theirs was essentially a symbolic role, projecting authority, power, and wealth to those being ruled. - page 7

This quote also mentioned the first point made (which I mentioned we would revisit) as to why there was a difference in building material between Japanese castles (tenshu more specifically) and European castles, which is that the castle complex as a whole was what provided the defensive capabilities. In fact, there were numerous castles of the period that would have a complex of walls, ramparts, earthworks, bridges, etc, but would not have a tenshu (like we imagine today) as part of the complex. Benesch and Zwingenberg talk discuss how it was the elements surrounding the tenshu that provided defense when stating:

...tenshu, are but one part of the larger castle, which may also have original moats, stone walls, earthworks, gates, bridges, and other features. In practical terms, these latter structures were more essential to castles' capabilities as military fortifications, and many castles did not originally have tenshu. - page 7