Is the word serviço the same in Brazilian and European Portuguese?

No.

While the fundamental definition of "serviço" (service, task, or duty) is the same in both dialects, there is a significant difference in typical usage. In Brazil, "serviço" is very commonly used as a synonym for "trabalho" (work) or "emprego" (job/employment), particularly in informal or semi-formal contexts. In Portugal, while "serviço" is used to describe a specific task or a professional service (like "customer service"), it is not typically used to refer to one's occupation or employment; a Portuguese person would much more naturally use "trabalho" or "emprego" to express that idea.

Brazilian Portuguese Usage

1. Estou à procura de um serviço novo. (I am looking for a new job.)
2. Esse serviço ficou muito bem feito! (That job/work turned out very well!)
3. O serviço de quarto do hotel é excelente. (The hotel's room service is excellent.)
4. Ele faz um serviço de confiança. (He does trustworthy work.)
5. O serviço de entrega atrasou hoje. (The delivery service is late today.)

Continental Portuguese Usage

1. Estou à procura de um emprego novo. (I am looking for a new job.)
2. Esse trabalho ficou muito bem feito! (That work/job turned out very well!)
3. O serviço de quarto do hotel é excelente. (The hotel's room service is excellent.)
4. Ele faz um trabalho de confiança. (He does trustworthy work.)
5. O serviço de entregas atrasou hoje. (The delivery service is late today.)