Ostrich milk was, until the mid-1800s, an alternate name for ostrich oil, an oil made from rendered ostrich fat that is said to be very good for the skin.
At some point the language shifted, and the older usage remained as an occasionally glimpsed but rarely understood non sequitur. Unwilling to let an odd phrase go to waste, people started using ostrich milk as a silly or unobtainable item, much akin to hen's teeth, mare's nests, or a left-handed monkey wrench. This is still the most common and most cromulent use of the term.
There are reports of people harvesting the crop milk of ostriches to serve to tourists, although as it happens ostriches do not produce anything of the sort. (Ostrich chicks do gain some antibodies to support immune functioning through the consumption of parents' feces. This is in no way related.) Whether it is the milk or the reports that have been faked, I cannot say.
And finally, many various things have been named ostrich milk just for fun, the most common of which appears to be variants of horchata mixed with alcohol.
Brevity Quest 2016