In my experience as a native speaker of American English, the date is sometimes spoken as all numbers, particularly when reading a date written that way. However, it's in the format "two thirteen", where the first number is the month and the second number is the day. In addition, sometimes a third number (the year) is included, either by saying the entire thing ("two thousand and five") or the last two digits (e.g. "oh five"). And sometimes "dash" is said as a separator, since it's pretty common to see dates written as "9-21-18".
Due to the nature of search engines, it's a little hard to find evidence online to back this up. I was however able to find a few examples (although they are mostly written):
Some special dates are likely to be referred to in this fashion. The example that comes right to mind is 9/11, which is almost always referred to as "nine eleven". And then there's "eleven eleven", which the one year was "eleven eleven eleven". And, as 1006a mentions in a comment, Pi Day is called "three fourteen". Depending on who you talk to, there's also four twenty. However, the Fourth of July isn't called "seven four".
There's another similar way to describe a date using numbers, but it's only used in "poetic" contexts. The examples I know of for this are:
- "The eleventh hour of the eleventh day of the eleventh month" is when the Armistice of Compiègne was signed.
- Various Bible translations have "the eleventh month of the twelfth year, on the first day of the month" in Ezekiel 26:1, but this is a different calendar (the New Living Translation has it as "February 3"). It's used in some other places in the Bible too.