The sun (due to earth's rotation around the sun) returns to the same relative position approximately every 365 d 5 h 49 m 30s.
This period is not divisible by a whole day. So the sun will rise marginally off the same direction at any given location in one year. This being marginal, let it be ignored for the current discussion.
So the sun will rise from the same direction at any given local every year on the same day of the year.
The best way to determine the same day of the year, is the number of the days this day occurs after the day in which say Spring equinox is observed.
An alternate way of determining the same day of the year is to look it up on the Gregorian calendar.
The latter way of determination of the same day of the year could be off at some locals by one day.
The main reason for this is that the calculations for the Gregorian calendar are based per locale and if applied globally
Without altering the dateline to the longitude where It is midnight at start of the year will result in one day of error (difference in date of the same day of the year determination from the former method, which is the number of days counted since the equinox.