Normally, time periods don't have a profound influence in shonen battle anime. Whether a series takes place in the 21st century or the 19th century won't really change how one character punches another, especially if special powers are involved. With that said, there are ways a series' time period can be seen and felt. A good example of this is Demon Slayer, the unique time period of which is exceptionally noticeable.
Demon Slayer takes place in the Taishō Period, which is between 1912 and 1926 in Japan. This period in time is highly distinguishable by its adaption of technology, its fashion choices, and its clash of traditionalism and modernism. Not all of these things figure into Demon Slayer fights, but they are still palpable throughout the series' narrative.
One of the most notable ways the time period can be felt is through the technological advances of the period. Coming off the Meiji Period (1868-1912), Japan's efforts to modernize had officially borne fruit in many different ways. In Demon Slayer, this is initially observed through things like power lines in the rural areas; as bigger cities are explored, things like street lamps and automobiles become standard. There's an entire arc that revolves around the main cast riding a train; this wouldn't mean much in most series, but in Demon Slayer, it's a rare experience. These are all things for characters like Tanjiro and Inosuke to marvel at since they grew up in the countryside and are thus unfamiliar with many of these advancements.
That difference between rural and urban areas is also seen in how people treat demons. In rural areas, which still adhere to tradition, demons are seen with fear. this is why people try to stay indoors at night.
Urban areas are less afraid of demons. People regularly walk around at night in large groups with no sense of danger. This is a bold lifestyle choice, but it's also what makes it so easy for Kibutsuji Muzan to casually walk up to somebody, slash them, and turn them into a demon.
In any case, it's obvious that demons and other supernatural elements are being forgotten. As science progresses and the unexplainable becomes explainable, fantastical things like immortal demons with magical powers become harder and harder to believe. Even the people of the countryside only view demons as rumors and superstition. As Japan moves further into the modern age, they forget to fear demons or that they ever even existed.
Another way Japan tried to modernize in the Taishō Era was through its adherence to western fashion. The age of kimonos and zōri sandals was coming to an end, which made room for three-piece suits and penny loafers. These kinds of changes with the times are observable even in older characters like Muzan and Enmu. It can also be seen in the Demon Slayer Corps' gakuran-like uniform, an outfit that finds its roots in Prussian Waffenrocks; many of them also tend to wear a haori jacket over it. There are all sorts of characters and background characters who help represent this transitional age in Japanese fashion.
The Taishō Period also continues to enforce the sword ban of the Meiji Period. With the end of the Warring States Period, swords were only allowed to be used by a select few, particularly government officers. This ends up being a major hindrance to Tanjiro and his friends as they must conceal their own swords even when they're on official Demon Slayer business. It might have been more convenient to simply set the story in an earlier era, but having everything happen after swords were banned adds to the clandestine nature of the Demon Slayer Corp.
Unlike most shonen series, Demon Slayer's setting has a number of profound effects. For one thing, since the Taishō Era is rarely covered in such great detail, not many series look quite like Demon Slayer, except maybe Rurouni Kenshin. Beyond aesthetics, the Taishō Period also contextualizes a lot of the characters' words and actions. It gives the series a unique look and feel as well as key plot development. All of this makes Demon Slayer's setting refreshing and engaging.
