Why did neither Japan or China become a democracy until the 20th century?

I corrected your question; Japan did become a democracy in the mid 20th century, as a result of the US occupation after WW2. Technically they are a constitutional monarchy with a parliamentary system---but so are the UK and Sweden, which we widely recognize as democratic states. Japan does have what's called a dominant-party system where a single political party (the Liberal Democratic Party) generally wins most of the seats; but as far as we can tell this isn't because the government of Japan represses other parties, it's simply because the Liberal Democratic Party is extremely popular and keeps winning the most votes.

China really still hasn't become a democracy, however; while they do hold elections, these elections are tightly controlled by the government and limited to Communist Party candidates. Their human rights record is quite bad, they censor most media heavily, and they have been criticized by the UN for holding sham elections instead of establishing a true democracy.

Of course, the question you asked is why---why did Japan take so long to establish democracy, and why has China still not?

One theory is that there are some sort of "Asian values"---cultural norms in Asia, particularly in China, that teach people to think in terms of the community rather than themselves, sometimes called communitarian as opposed to individualist values. These values seem to be reinforced by the major religions of the region, namely Buddhism (in both countries), Confucianism (in China) and Shinto (in Japan).

Yet this theory is not all that well-supported; nobody has been able to clearly define and document these alleged differences in cultural values.

But in fact, I think the question is backward: It is not hard to explain why Asian countries didn't become democratic---it is hard to explain why European countries did.

Even Europe didn't really have all that much democracy until the end of 18th century. While some countries had constitutional monarchies that gradually expanded power for their people (particularly the UK and Nordic countries such as Sweden and Denmark), no country in Europe really had a democracy in the modern sense until after the French Revolution (which didn't really end until 1799).

Living in First World countries today, we often think of democracy as though it is the obvious default, but in fact it is quite aberrant. Monarchy, oligarchy, theocracy, and dictatorship are much more common systems of government throughout history. Indeed, true universal suffrage was not present even in the US until the 20th century---specifically 1920 and the 19th Amendment granting women the right to vote. With this in mind, it should not seem so baffling that Japan did not have a democracy until 1945; that's only one generation behind.

The reason Asia did not democratize quickly can then be explained by the fact that democracy spread like a contagion from its roots in the United States and France, first changing countries that were nearby and culturally similar, and only later spreading to most distant and culturally different countries via trade (and warfare). If this theory is right, then we should expect democracy to continue to spread as it shows itself to be a superior system, perhaps eventually taking hold around the world. We can hope, anyway.