To help conceal his identity, Superman (who has a birthday on February 29th), doesn't share a birthday with Clark Kent (who was born on June 18th!)!
You can see, that DC has done a few tricks over the years to put Superman at odds with Clark Kent...
One of the biggest tricks they pulled was having Superman and Clark Kent team up, and they did that twice in Superman's team-up book of DC Comics Presents!
Superman meets Clark Kent
Eventually, Superman and Clark worked together and returned to being one man.
Clark Kent Meets Superman
It might seem strange to devote time to Superman and Clark Kent working together, but they need to, and sometimes we all need to make parts of ourselves work together...the showy larger than life side full of action and heroics (one might say the Superman side) and the quieter, thoughtful, more emotional reporting side (the reporter, Clark Kent) to make us all be one functioning person!
...and, oddly enough, in 1908, it was when Bud Collyer was born (he who would voice Superman on radio and in the cartoons of the 1940s, shifting his voice an octave when he switched identities).