Question on Levinson's proposed discrete form of Wiener filter
Not specific to statistical Wiener filtering but also true in general, any algorithm will fail to work, if its assumptions are not met. However, it's also known that most algorithms will perform quite acceptable when their assumptions are sufficiently holding.

In this sense, if a statistical algorithm relies on the stationarity of its inputs, then it will be acceptably performing fine when the inputs are sufficiently stationary. Hence the keyword is sufficiency...

One might therefore argue that, a slowly changing (in statistical characteristics) non-stationary input would still allow the short term predictions possible , albeit with more erros than the ideal expectation of a strictly stationary input. 

Taken the other way, many (if not all) nonstationary processes can be considered to be approximately stationary over a short enough period of time. Indeed this is the whole basis of applicability of most statistical signal processing algorithms that makes any use of real world data (which is, as you have outlined, nonstationary by its very nature, except the exceptional cases). Divide the input into pieces, assume stationarity over that period, apply the algorithm.

That being said for certain engineering signals of the type: speech, music, image, video, radar, sonar, seismology, you should make it clear by yourself that those financial data are also approximately stationary over some short periods of time. Otherwise you will have trouble in applying those algorithms that rely on stationary inputs to work.