I was just wondering because both movies star him as an astronaut trapped on an isolated planet. They also had fairly close release dates. I am wondering if there is a link between the two characters and if the movies were supposed to tie into each other.
Firstly, The Martian was originally a novel released chapter by chapter by author Andy Weir on his website and later edited and published as a novel. The novel inspired a film starring Matt Damon. The book was officially published in 2011, 3 years before Interstellar. Therefore the premise of your question is incorrect.
As for the possibility of them being the same person (besides the fact they have distinct names)
Mark Watney arrived on Mars in November 2035, however Cooper from interstellar, who Dr. Mann was of similar age to, was born in 2031/2032. The wormhole only having been detected in 2019, with the events taking place 48 years after it's discovery (Dr. Mann being shipped off 38 years after).
For the curious: Ares 3 launched on July 7, 2035. They landed on Mars (Sol 1) on November 7, 2035. The story begins on Sol 6, which is November 12, 2035.
Andy Weir post - Facebook
2031/2032: Cooper is born in Denver, Colorado.
Timeline of Interstellar - Interstellar wiki
Therefore, Mark Watney would've been landing on Mars at around the same time Dr. Mann was born. The timelines don't seem to add up, Mark Watney would've been at least in his 50s/60s at the time he shipped off on the Lazarus missions.
So Mr. Watney,” the boy said, “If you could go to Mars again, like, if there was another mission, and they wanted you to go, would you go?”
Watney scowled at him. “You out of your fucking mind?”
The Martian - Original Ending as recorded on The Martian wiki