Is it obligatory to wash clothes suspected of being contaminated with madhy (pre-seminal fluid), and what is the ruling on performing ghusl (ritual bath) afterward, given that one does not know the appearance of madhy?

There is no need for distress, for the matter is merely a whisper (waswasa), and you must disregard it. What you saw might have been normal vaginal discharge, which is pure, so there is no need to purify clothes from it. And if it was madhy (pre-seminal fluid), then your mother washing the clothes in the washing machine is sufficient to purify them. If you are in doubt as to whether it is madhy or vaginal discharge, then choose and assign to it the ruling you prefer.