Regression over the whole population
It's important to consider what exactly the population is about which an inference is being drawn.  It's easy to overlook the time aspect in this context.

Suppose for example that the aim is to forecast the next two years' GDP for each country in the world. Then the population of interest is a set of pairs of the form "country, year".  It isn't simply "all countries", and even if a forecast model has been estimated by regression on current and past years' data for each country, that doesn't mean that the whole population of interest has been included.

If one really does start from a complete dataset for the whole population of interest, then all one can do is calculate summary statistics. That could include standard deviations, but it would be inappropriate to call these standard errors, since that term relates to a sampling distribution whereas the only "sample" in this case is the whole population.