Is it obligatory for me to purify myself from madhy after every urination, even if I don't see it, due to the possibility of its discharge with urine? And is this act considered a form of waswasa (religious scrupulosity/obsessive thoughts) or a precautionary measure?

Purifying madhy (prostatic fluid) is similar to purifying urine according to the majority of scholars: whatever it touches on the body should be washed, and it suffices to sprinkle water on clothes. Only the exit point of the impurity needs to be washed. If you are certain that madhy has exited, wash whatever it touched on your body, sprinkle water on your clothes, and perform ablution (wudu'). If you doubt its exit, nothing is required of you. If the discharge is urine, perform istinja' (washing the private parts) for it, and you are not required to wash the entire penis and testicles. Beware of obsessive whispers (waswas) and yielding to them.