US: not really, no. People will understand you if you say "I was born in the fifth month of the year 2003." But it sounds overly formal. It is the type of thing that you might say if you were a politician giving a very, very important speech. In most situations, we would say "May."

Using ordinal numbers to refer to months isn't the most common way, but it is not unheard of. When someone asks me my birthday, I often reply with 'twelfth of the twelfth' - i.e. 12 Dec. I have no idea why I say that (I grey up in England and am now Australian) but I have never had anyone misunderstand me.

Angela's answer would usually be right for spoken British English as well. However, there is one special context which is different. When people write a date quickly, they might write it as "1/11/18". People would usually pronounce this as "the first of November twenty-eighteen". However, in a situation with limited time (like saying your birthday to the bank), some users of UK English would say "first of the eleventh eighteen". I think it's especially common when this is quicker than saying the month's name ("first" has one syllable, "January" has three), though I'm not certain. Can any Aussies, Indians, Singaporeans, etc., tell us whether this happens with other forms of British English? I can also tell you that in Mandarin... well, look at the words for yourself: 一月 二月 三月 Even if you've never studied any Chinese, you might have guessed that the the names of the months are simply "Month One", "Month Two", etc.