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

Yes

The meaning, grammar, and spelling of "reservar" are identical in both Brazilian and Continental Portuguese. The only differences lie in pronunciation. In Brazil, vowels are generally more open, and the "r" at the end of the word is often softened or nearly silent in many regional dialects. In Portugal, vowels are more closed and "reduced," and the "r" is typically more pronounced or aspirated.

Brazilian Portuguese Examples

1. Eu gostaria de reservar uma mesa para duas pessoas. (I would like to reserve a table for two people.)
2. Vou reservar um quarto no hotel para o fim de semana. (I'm going to book a room in the hotel for the weekend.)
3. Precisamos reservar as passagens aéreas o quanto antes. (We need to book the airline tickets as soon as possible.)
4. Você pode reservar uma sala de reunião para nós? (Can you reserve a meeting room for us?)
5. Ela reservou um tempo especial para estudar. (She set aside a special time to study.)

Continental Portuguese Examples

1. Gostaria de reservar uma mesa para dois, por favor. (I would like to reserve a table for two, please.)
2. Vou reservar um quarto de hotel para o próximo mês. (I'm going to book a hotel room for next month.)
3. Temos de reservar os bilhetes de comboio com antecedência. (We have to book the train tickets in advance.)
4. Pode reservar uma sala de reuniões para nós? (Can you reserve a meeting room for us?)
5. Ele reservou um lugar para mim no carro. (He reserved a seat for me in the car.)