Upon further investigation, I must conclude that Shift JIS is a misnomer. Rather, this is codepage 932. Unicode and Microsoft provide a mapping table between this and Unicode. This is apparently what is being used to map the characters. Notice that it does not contain a mapping between (0x5C, U+00A5) and (0x7E, U+203E).
Note though that I wrote in the original question that "I have verified that while the yen sign (U+00A5) properly becomes 0x5C". Apparently, the Encoding.GetEncoding(String) method returns an encoding which has a DecoderFallback defined as System.Text.InternalDecoderBestFitFallback, which I assume is providing additional mappings for some characters which would normally fail. It must contain an additional mapping for yen (U+00A5), but unfortunately nothing for overline (U+203E). When I replace this with EncoderExceptionFallback if fails for bother characters.
Hence, I conclude that for Shift JIS, this is an error. But for codepage 932, it is the expected result.