Why couldn't the Ottoman Empire modernise at the same time as Meiji Japan did?

I would challenge the initial assertion that Ottoman Empire couldn't modernize, or that its demise was caused by an inability to modernize. The empire was defeated in WWI and consequently partitioned. But the seed of that destruction was planted decades before. https://en.wikipedia.org/wiki/Ottoman_wars_in_Europe#Ottoman_decline_(1828%E2%80%931908) Just look at the number of wars it got itself into in its final 100 years!

But in 1914, the Ottoman Empire was still a formidable near peer rival to her European neighbors, albeit a bit disadvantaged. It pretty much held its ground on its own (with some German support) against the full might of the Entente for nearly a year at Gallipoli, and ultimately won! Had the German Empire won in the West, we would be talking about the Ottoman revival of the early 1900s.

Meji Restoration took place in the late 1800s (1860s+) and transformed Japan from a mostly agrarian feudal society into a modern one.

My assertion would be the Ottoman empire was not nearly as far behind the development curve in the 1800s as Japan was. It was just disadvantaged by distance between it and Europe, making the Ottoman easy picking.

But there seems to be a second component to the question: How did Japan carry out its transformation so quickly?

The answer is clearly cultural. Every East Asian Confucian based society was able to rapidly transform itself, when the conditions for transformation were finally met (the end of occupation, or extreme political oppression)

Japan in the second half of the 1800s, was the first to make the move. Followed by Hong Kong (post WWII in the late 40s), S. Korea (post Korean war in 1953) Singapore in post independence from Muslim Malaysia (1965), Taiwan (post one party rule) and China (post 1980s).

And every one of them now is either a peer or near peer society of the West, catching up on nearly 500 years of progress in around 50.

No other region or cultural groups have managed anything like it.

And as an outsider historian and observer, it seems the key isn't democracy as we in the West would like to think. Singapore is not democratic. China is definitely not democratic. Even Japan lives under one party rule, with the LDP being overwhelmingly dominant. The difference makers seem to be the dedication to education, the adherence to a civic code of conduct, the penchant for saving, and the lack of appetite for adventurous wars.

All these nations haven't been involved in any serious military conflicts for years. (Japan since 1945. S. Korea, since 1953. Singapore since its founding in 1965. Taiwan since 1949. China since 1979) allowing their people to focus on building and investing in the future.

It is very hard to answer in depth such a broad question. Only general suggestions can be made I think.

The similarity between the two is limited: Japan was not a multi-ethnic, multi-religious empire extending on three continents.

The Japanese institutional unity, robustness and coherence was arguably greater, no matter the feudalistic aspects, which were present in the Ottoman empire anyway, and which partially remained typical for Japan no matter the modernisation.

Japan had followed an intentional and systematic isolationist policy during Tokugawa period between 1603 and 1868; that is, its isolationism was conscious, controlled, political, intentional (as well as geographical), rather than cultural and accidental; that was decided in a logic of competition with the West that included from the beginning the possibility of switching between the two options of isolation vs. integration; for the Ottoman the process was different.

When the evidence of Western technological superiority became obvious in the nineteenth century, Japan was thus able to reverse its course in contradiction with the previous stance, but based on reasons that were not alien to the previous course of action. The Ottoman had never been confronted with that logic, they never tried to avoid integration, they had been an expansionist Islamic empire with no national identity in the nineteenth century sense. The reform in its case meant more than choosing between two options, but represented an in-depth transformation that involved the confrontation with a crisis of identity.

I think the problem of crisis of identity stays as the greatest difference between the two, involving mainly the national/nationalistic aspect.

The comparison between the two may be tempting now, that is between post-Atatürk Turkey and Japan; but Turkey is the result of the collapse of the Ottoman, and the violent invention of a national identity (in the context of a World War and even of a civil war that involved ethnic cleansing and extermination) that in Japan was taken for granted.

And there is also the religious and cultural aspect which is even more complicated if not impossible to explain, concerning mostly the capacity of Japan to adapt to the industrial era. That is suggested in other answer(s) already posted here, but the "essence" or cause of that capacity is very debatable. Religious aspects can be put forward, but it is very hard to be sure they are decisive. I would mention for the sake of contrast Emmanuel Todd's anthropological theory that correlates Japan's (as well as Germany's) industrial success (and authoritarianism) to a specific type of basic family structure.

But I am sure many other such factors can be brought here into play.