In some ways this is all about random convention, but in the spirit of making this be about English grammar, notice that nineteenth is an ordinal while nineties is not.
That is, having a nineteenth century presupposes an eighteenth, seventeenth, etcetera, on back to the first century. The first century of the common era is defined as the years 1-99 (or 100 for the extreme pedants among us). The reason for this is probably best left as a question for History SE.
The criterion for the term "nineties" is different: it only has to have "ninety" in the number. Nowadays, this usually refers to the decade 1990-1999, but the term "gay nineties" used to be quite common, referring to social and cultural events of the years 1890-1899.
Although we now use "nineties" to refer to a decade in some century or another, it wouldn't surprise me(*) if some 11th-millennium wag would refer to the years 9000-9999 nostalgically as the "nineties".