So I was confused about your question because I am familiar with European castrati who were used as singers. The castration process for them only included the testes. They would either entirely remove them, crush them to uselessness, or just sever the spermatic cords. This doesn’t generally include the penis itself, so they would still be able to pee.

China has a long history with castration, though up until the Ming Dynasty it was reserved mostly as punishment for prisoners of war or criminals. The Ming Dynasty eunuch, which was generally performed voluntarily, required removal of both the testes and penis, and castration of this kind can also be referred to as emasculation.

According to my research, it was done with a curved knife, and they would try to remove both organs in a single cut. A needle or spigot was placed in the urethra, the entire wound dressed in wet paper, and the new eunuch was walked around for three hours before they were allowed to lie down. They were prevented from urinating and hydrating for three days, after which the spigot was removed. According to sources, if urine flowed out afterward, the operation was considered successful. Apparently only 2% of cases performed this way failed and caused infection and death.

This operation caused the eunuchs to have difficulties controlling their bladders, and they often wet themselves, giving rise to the aphorism “as smelly as a eunuch.”

It’s pretty gristly research. I found this information here: https://www.inkstonenews.com/china/how-men-became-eunuchs-serve-imperial-chinas-forbidden-city/article/2179649.

Modern castration is mostly done through chemical means, requiring regular injections of anti-androgens which render the testes inert.