This exists in Hebrew as well. The word for "hello", "שלום", also means "peace". "You're welcome" is "על לא דבר" which, like the examples above, means "for nothing", implying much the same. One of the words for "goodbye", "להתראות", means "for meeting again", similar to "see you" in English. This seems to be shared by both eastern and western languages by what we've seen so far. I wonder if some more esoteric languages have constructs similar to these. I don't want to jump to conclusions, and my knowledge of the subject is far from expert, but the impression I'm getting is that this is someting innate in human language.
Huh, that's interesting to use "peace" as a greeting. Every time I've used it it's always been as a farewell. You might be onto something there. I too don't claim to be an expert, but but if over time, these phrases began to resemble one another by demand. In order for tribes of people with a specific type of good to trade, they had to communicate in some sort of way, right? Again, no expert, just thinking about interesting possibilities.