The answer is...they were not.
The reason Sekigahara and Osaka were so big is because they were not battles between two armies, but battles between I-don't-know-how-many armies. Seriously. I don't. Like Shikoku, Kyūshū, and Odawara campaigns before them, they involved (semi-)independent forces from large areas of Japan. It was due to circumstances, political as well as military, these forces were concentrated on one battlefield for battle. Imagine if half, or even a quarter, of the armed forces of the Spanish Empire suddenly decided to concentrate at Madrid for a show down, well that would be a gigantic battle as well. Individual daimyos usually can only mobilize in the thousands (the smallest daimyos not even), and only the largest of daimyos could mobilize more than 20,000. Ieyasu, by far the largest daimyo present at Sekigahara, brought only 20,000 to 30,000 men. Now he did hope to have 40,000 to 60,000 men, but on a military decision he sent his son off with a separate army, and his son didn't arrive in time.
As explained here, the Japanese did not mobilize a larger percent of its population for war than Europe. Or, at least, some European polities mobilized as much, or more, men as a percentage of its population compared to Japan. So it's not like European states did not have the ability to engage in such gigantic battles. There was just no need or point in doing so.
If you go through the engagements in the Korean campaign, which politically and militarily should be more in line with the type of wars Europe was fighting, you can see the number for Japanese forces are brought more-or-less in line with Europe.
OP, do you have a source on the hundreds of thousands claim? That sounds to me like an exaggeration, but I don't know much about Japanese history.
Related question, how would one move so much food and water for those armies?
Related question: what happened in Japan immediately before the Sengoku period in the early 15th century? Was it a relatively peaceful time after the power struggles of the mid-14th century?
If so, returning to OPs question, could the difference simply be due to available fighting population? England by this time, for example, had experienced the Great Famine at the beginning of the 14th century, followed by the plague, the Hundred Years War, and then the Wars of the Roses. By the time you get to the end of the 15th century, did they really have enough people to muster such large armies? For comparison, a cursory online search suggests that the population of Japan in 1450 was ~10 million while the population of England was below 3 million.
This strays a bit from the topic, but it's something i've thought about and wanted clarification of and covers a similar subject: If you do reading on Mesoamerican warefare online, you'll not-uncommonly see numbers in the hundreds of thousands for army sizes. Even here on Askhistorians, I see flaired users refer to these in instances with no implication of them being inaccurate, while other times other flaired users do use lower estimates for the same things or concede these are likely to be hyperbolized, but the only specific numbers I've consistrntly seen users here note is likely to be exaggerated is the report of 400,000 Aztec soliders taking Tututepec.
I thought maybe a potential explanation here is that the 100,000-200,000+ army numbers included noncombatants that were transporting supplies, since the region had no beasts of burden to assist in that regard, but then I also often see that the numbers list hunderds of thousands of soldiers plus tens or 100,000 porters.
What's the deal?
Any Chinese historian can comment on the same question but regarding individual states during the warring states period? Even during war between 2 out of several minor states that eventually make up current day China, there are still battles with armies numbering the hundreds of thousands. Alexander manage to conquered most of the civilized world with a fraction of those numbers and then Rome manage to be the dominant force in Europe for hundreds of years with much less.