> What I like about it is that it arranges things in order of importance/relevance in most situations. The month tells me the most important bucket, [...]

In my experience, the thing which is the most important/relevant in most situations is the day of month. People often say things like "dia 10" when asked when something is going to happen; that means in context the 10th of the current month if today is the 9th or earlier, or the 10th of the next month if today is the 11th or later. The month is added only when necessary, like "dia 10 de fevereiro" (just "dia 10" today would mean the 10th of January).