Is it obligatory to repeat the Fajr prayer after feeling wetness or discharge and doubting one's state of purity?

You are afflicted with Waswasah (obsessive whisperings), and the cure for it is to ignore it and pay no attention to it. If you are not certain that something was emitted after performing ablution, then your prayer is valid and you are not required to repeat it, because the default state is the continuation of purity, and certainty is not removed by doubt. What you have done by ignoring the whisperings is the correct approach.