u/godisanelectricolive answered a similar question last year:

Australia, New Zealand and Canada all switched from currencies called "Pound" to currencies called "Dollar" in the 20th Century. Was this to gain diplomatic and economic favours from the USA or was there some other reason?

More can always be said, but this rather old answer of mine talks about why the Canadian currency is the dollar.

While the other answers address the British settler colonies well, another point to consider is the sheer scale on which the Spanish (or later Mexican) silver dollar circulated as the primary or secondary minted coin throughout the world, especially in maritime Asia, in some places into the late 19th century. There are a couple of supply side factors contributing to this: the massive quantity of silver extracted from the Spanish Americas, the quality of their minting processes, and the global expanse of the early modern Spanish/Portuguese empires. This is compounded by the tendency for many Asian states, especially the Ming and Qing empires, to not mint their own silver currency. Since silver coins of known (and steady) silver content are more convenient than raw bars or potentially dodgy private mintings, Spanish dollars were a key part of Asian maritime and inland trade networks. They would have been more common in coastal areas exposed to world trade but I have seen them in court cases as far upriver as 19th century Sichuan.

As state building in the 19th century progressed and Asian states sought to exert more influence over their own monetary systems, they usually created national currencies (with varying degrees of success) that usually borrowed older terms for currency and added a "national" modifier. The best example is the Singapore dollar - this was a place where despite nominally belonging to a part if the British empire where the rupee was the government unit of account for most of its early history, Spanish dollars were always the most important actual coin used, and thus became the naming basis for the later fiat currency. In the PRC, the yuan renminbi actually followed a similar trajectory: in Qing documents the term "yuan" 圓/元 (lit. Round) usually refers to minted silver coins and frequently specifically to Spanish dollars. Thus you could conceivably translate yuan renminbi as "one dollar of the people's currency."