This may not be what you had in mind, since I can only speak for Italy, but here it goes anyways (and with the additional observation that, given the importance of cultural phenomenons for the use, things may be very different in other countries).

In terms of the abbreviation, they did. The battle of Adwa was the one of '96 (the whole conflict being at times called "the war of '96") and the class of 1899 - those eighteen years old conscripted after the defeat of Caporetto in 1917 - came to be known as "the boys of '99". More so the '48 was obviously the year of the great European revolutions, 1848 - and the expression, "to make a '48" remained in use to design a great upheaval or just a big mess.

Above all though, the choice to abbreviate the year was driven by the assumption that the public was going to understand it. Here for example Mussolini asks a rhetorical question (July 3^rd 1919):

Should we assume that the Combatants Association is a conclave of reactionaries from '98?

And he obviously means 1798, that is the first period of restoration after the establishment of the "French" republics in Italy during 1796-97.

This should also explain why references to 80s and 90s were somewhat less ubiquitous than they are now, as they appear to represent for us not only the abbreviation of a specific year or decade - even if they could and were used that way at times - but to identify a "cultural" period. In this sense, more often than not, the Italian public referred to events or periods in a way that was both characteristic and understandable, for them. The 1890s were split in half by the end of Crispi's rule, the aforementioned defeat of Adwa, the end of the Fasci Siciliani, going from a period of attempted national expansion to the attempts at authoritarian rule under Rudinì and Pelloux. There was little reason to look back at those years as a unified thing, but there were references to "the years of the Fasci", meaning somewhere around 1893-95 but in general just referring to the period of social tension and police and military repression (the reference of course would carry a different meaning in Sicily or Piedmont). The decade 1901-11, which one could find shortened into "the nine hundreds", was well known from the name of recurring Prime Minister Giovanni Giolitti, even if the expression "Giolitti's Age" found its complete affirmation only later on; similarly the decade after the unification was the "age of the right" from the party of government, later transformed into "historical right" to distinguish it from the "national right" of early XX Century.

In other words, the decades themselves didn't appear to serve as cultural identifiers, but were used for shortening when the meaning was obvious. "Back in the '90s" would be understood as 1890s according to context, but it probably wouldn't carry the same significance we attribute to the '60s or '70s nowadays.