It's certainly true in spoken language both orders are present, also over here, but there are lots of things that are always done following a fixed way in writing and however suits the speaker and the occasion when spoken aloud. Spoken language is usually less formal than written language (in Finland exceedingly so), and I personally think it's better to consider the written language as the norm, not the other way around, partially because the spoken language changes faster and varies more between regions.

In short, even if you usually say it month first, day second, doesn't mean it should be written in that order as well. However, I don't expect it to ever change. After all, you never got rid of the feet, miles, and pounds either.