During the 1860s, the Tokugawa shogunate paid a large indemnity to European powers in Mexican silver dollars: a detail that I finally got curious about the other day. Why Mexican silver dollars? Googling tells me that Mexican silver dollars were everywhere in East Asia, and had become a preferred form of currency in China by at least the 19th century. In Japan, they were stamped with the shogunate's determined value and sent into internal circulation.
Why would these coins be so wide-spread and in such demand for commerce? How did they get in such large quantities to East Asia? I have a few hunches, but would like to hear real information.
From the 16th century onwards, Spain had both control of silver mines in Peru and Mexico as well as control of much of the Philippines, which were the main entrepôt for Spanish commerce with East Asia.
The so called Manila galleons transported Spanish silver to the Philippines so that Spanish merchants could use it to pay for East Asian silks, ceramics, etc. This trade continued over centuries, so it certainly makes perfect sense that the Asian economy was able to accumulate a significant surplus of Spanish silver. The early modern period (16th-17th centuries) was also one in which Europe had little to offer the 'Far East' in exchange for Asian manufactures and spices. One of the few things Europeans could provide (and which would survive the long trek East) was precious metals. Slowly, over the centuries, exceptions developed (textiles, guns, etc.) and eventually the paradigm changed completely, with European products dominating many Asian markets, but this was only a later development, dating from the late 18th and 19th centuries.
Keep in mind that the amount of precious metals extracted from the Americas was so large that it helped revolutionize the entire European economy at the time. Gold and silver served both as currency for circulation and as material reserves for banking and finance. So much was brought to Spain that it caused massive inflation in Spain itself and was able to help 'lubricate', as it were, the economies of the rest of Europe.
For just an idea of how much silver we're talking about during this period, speaking solely of the Potosí mines in Peru, "Potosí produced some 22,695 metric tons of silver" between the years 1545 and 1823. There were others mines both in Peru and, especially later, in Mexico, which provided massive amounts of gold and silver. This doesn't even count the influx of precious metals from Portuguese Brazil, British South Africa, or other colonies around the world thanks to mining.
It also isn't too surprising that the 19th century sees large amounts of the silver Asian state's had accumulated earlier find its way back to European banks. By the late 19th century, many Asian economies were dealing with significant trade deficits with European powers that were bleeding them of precious metals. The most famous case, British exports of opium to China, led to the so-called Opium Wars between the two States. In the case of India, Sven Beckert points out in Empire of Cotton how a flood of cheap British textiles, and other factors, eventually led to the country becoming a net exporter of raw cottons and a net importer of finished textiles. In the specific case you cite, the military superiority of Western powers also helped extract precious metals from Asian economies.
In short, the superiority of many Asian manufacturing industries that had lasted millennia was reversed and the flood of bullion from the Americas slowed, stopped, or (at times) reversed.
I'm less familiar with the specific case of Japan, but given the role of Spanish silver in the Modern economy, it is certainly not surprising.
Henry Kamen: Empire: How Spain Became a World Power, 1492-1763
Sven Beckert: Empire of Cotton: A Global History
Immanuel Wallerstein: The Modern World-System, parts I-IV.