> I’ll be the first person to admit I have no idea what postmodernism actually means

Well, if we want to take the title literally:

Error handling is a fundamentally Modernist idea - shoring up errors and "correcting" or at least containing them.

Postmodern error handling would be anything that reasonably counts as a reaction against Modernist error handling. The most basic would be not handling errors at all, and instead embracing the chaos of an error-prone system, and designing a system in which errors propagate but converge to a desirable value.