Is the Japanese phrase "俎板の上の鯉" - a carp laid on a chopping block related to the Vietnamese phrase "như cá nằm trên thớt" - like a fish on cutting board, either because Japanese got the phrase from Vietnamese, Vietnamese got the phrase from Japanese, or the two languages got it from a third language?
I don't have any clue to decide whether it's a parallel evolution or not, but I guess it's from Chinese, considering the phrase is attested in a famous (1st century BC) Classical Chinese literature, namely Shiji, and the fact all Chinese, Japanese, Korean and Vietnamese shares the similar expressions.
The most powerful never mind small matters; the most polite never care about small etiquette. Now they are knife and chopping block, we are fish in between, why need we say goodbye?
Korean: 도마에 오른 고기
Vietnamese: như cá nằm trên thớt
Chinese has 人为刀俎，我为鱼肉, which I must say matches the Vietnamese one more, and is well documented in its origins.
The phrase (in chinese) is attested to Sima Qian of the Western Han dynasty, who wrote in 史記・項羽本紀:
“如今人方為刀俎、我為魚肉。”, meaning "to be taken advantage of"
Many thanks to China documenting its history so thoroughly. Its structure doesn't match that of either the Vietnamese nor the Japanese, and because this is classical Chinese, I'd be rather surprised that the Japanese one didn't at least keep some of the characters (like 魚 vs 鯉) if this were the origin. Same goes for Vietnamese, which uses native vietnamese terms rather than Chinese ones.
Relationship between the phrases
If you look at the meanings of the two phrases in either language, note that the Vietnamese one is about your fate being out of your control (whether or not you like it), and the Japanese one is about staying calm in the face of an inevitable doom. They're similar in style, but different in ultimate meaning, which seems unusual if they were loaned. 一石二鳥 was loaned to English, and the meaning's still intact. I doubt in this sense that it's representative of being loaned just because they're both about fish being cut on a board.
The chinese expression doesn't really match the sentence structure of either of these at all, but it more closely matches the vietnamese one in meaning (not exactly). The Japanese one doesn't really resemble either in what it implies, which focuses more on the reaction of the person rather than the situation they're in.
If there was a common language, it's almost certainly chinese, but the chinese equivalent just mentions fish, and that's it -- not a whole lot in common. Also, the Viet and Japanese terms differ in meaning and style.
I'd therefore conclude they were developed independently by fishing-based societies. Their meaning, style, and structure aren't too much alike, but just resemble a common experience any fisherman has when cutting a fish.