I can't say much about other cultures, but in Eastern Europe, there was a concept of "Name Day" going back to Christianization of Russia. Typically, any day of the year had several saints associated with it. For example, November 19th is the name day for Alexander.

While people wouldn't necessarily be named following the name day calendar except in very religious families, they were generally aware of it, and would typically celebrate it, usually with a rich meal and close friends - that is, in more or less the same way we celebrate birthdays.

So someone born on November 19th, whatever their actual name, would often celebrate the name day of their patron saint.