Although I am not familiar with Taiwan, I have heard of similar practices in European countries as well. You may call it a 'tip', but I would rather call it a bribe. Even if you, or some sort of insurance, pay for your treatment to the hospital, it may actually be required for proper service to give an additional compensation directly to the doctor.

A quick search on Google reveals that a similar practice is well known in Taiwan as well:

https://www.sciencedirect.com/science/article/abs/pii/S0277953606004679

The Taiwanese practice of patients giving informal payments to physicians to secure services is deeply rooted in social and cultural factors. ...

Although I am not familiar with Taiwan, I have heard of similar practices in European countries as well. You may call it a 'tip', but I would rather call it a bribe. Even if you, or some sort of insurance, pay for your treatment to the hospital, it may actually be required for proper service to give an additional compensation directly to the doctor.

A quick search on Google reveals that a similar practice is well known in Taiwan as well. According to Gifts, bribes and solicitions: Print media and the social construction of informal payments to doctors in Taiwan:

The Taiwanese practice of patients giving informal payments to physicians to secure services is deeply rooted in social and cultural factors. ...