Similar pronunciations of tea/茶 across languages
Yep, they do
As you can find on the great and all knowing wiki, the word for tea in most languages comes from the Chinese:
Cognate to Min tê:

English tea
Danish, Icelandic, Norwegian, and Swedish te
Finnish and Estonian tee
French thé
German Tee
Greek τέϊον
Hebrew תה
Indonesian and Malay teh
A great many others

Cognate to Mandarin chá:

Greek τσάι
Japanese ちゃ and さ
Korean 차
Kurdish ça
Punjabi چا / ਚਾਹ
Portuguese chá
Turkish çay
And others

Cognate to Mandarin chá via Persian:

Arabic شاي
Czech čaj
Georgian ჩაი
Greek τσάι
Hindi चाय
Pashto چای
Persian چای
Romanian ceai
Russian чай
Turkish and Turkmen çay
Ukrainian чай
Urdu چائے
Others

There's some double counting going on here, but I'm just copying from their list.
Here's an interesting map showing the etymology of various words for &quot;tea&quot; in Europe.