This is so much desired when Discourse is used in technical contexts (e.g. Openwrt forum) and the posts dates matter a lot to quickly identify outdated content in long running threads! I would want to be able to change this from my user profile and not require each Discourse admin to change anything.
For the time being, I use the following bookmarklet to get full dates whenever I feel there’s ambiguity:
javascript:(function(){Array.prototype.forEach.call(document.getElementsByClassName("relative-date"), function(el) { el.innerText = el.title });})();
+1 for the suggestion about at least using full year, it is especially important for users in regions where year abbreviation is not a thing (France in my case).