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

No.

Differences in Usage While "garçonete" is understood in Portugal, it is not the typical way to refer to a waitress. In Brazil, "garçonete" is the standard, everyday term. In Continental Portuguese, the most natural and common term is "empregada de mesa" (or simply "empregada"). Additionally, while Brazilians typically use the word "cardápio" for a menu, Portuguese people more frequently use the word "menu."

Brazilian Portuguese Examples

1. A garçonete trouxe o cardápio. (The waitress brought the menu.)
2. Eu chamei a garçonete para pedir a conta. (I called the waitress to ask for the bill.)
3. A garçonete foi muito simpática conosco. (The waitress was very friendly to us.)
4. Precisamos falar com a garçonete. (We need to speak with the waitress.)
5. A garçonete anotou o nosso pedido. (The waitress wrote down our order.)

European Portuguese Examples

1. A empregada de mesa trouxe o menu. (The waitress brought the menu.)
2. Chamei a empregada para pedir a conta. (I called the waitress to ask for the bill.)
3. A empregada de mesa foi muito simpática connosco. (The waitress was very friendly to us.)
4. Precisamos falar com a empregada. (We need to speak with the waitress.)
5. A empregada de mesa anotou o nosso pedido. (The waitress wrote down our order.)