Why does nearly every word for tea sound like "tea" or "cha"?

Because the word reached the world from China along different trade routes, and each route carried a different Chinese pronunciation. Nearly all the words for tea worldwide fall into three groups, te, cha and chai, and the history of those words follows the history of the tea trade out of China. The "tea" group came from tê, the pronunciation in Min Chinese, picked up by Dutch traders either directly or through Malay teh. The "cha" group came from Cantonese and from northern Chinese, and the northern cha travelled overland along the Silk Road to Central Asia and Persia, where it picked up a Persian ending and became "chai".

English got two of them. The first to arrive was "cha", in the 1590s, via Portuguese traders in Macao who had picked up the Cantonese pronunciation. The more common "tea" came in the 17th century with the Dutch. So the tidy rule you may have heard, sea for "tea" and land for "chai", is only half right: the Portuguese brought "cha" by sea too.

A word that wanders this far from its source is called a Wanderwort, and tea is one of the standard examples. It is a kind of loanword, a word borrowed from one language into another, and loanwords rarely travel alone. After almost 350 years of Dutch presence in what is now Indonesia, roughly 20% of Indonesian words can be traced back to Dutch. Peter the Great studied shipbuilding in Zaandam and Amsterdam, and Russian kept Dutch words for sailor and jack. And borrowing can go strange places: the English word "Viking" became the Japanese baikingu, "buffet", because the first buffet restaurant in Japan, opened in 1958, was called "Viking".