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

No

While both varieties use "sobremesa" to refer to the sweet course at the end of a meal, there is a difference in meaning and usage. In Continental Portuguese, the word can also refer to the period of time spent at the table conversing with others after a meal has finished. In Brazilian Portuguese, the term is almost exclusively used to refer to the dessert itself.

Brazilian Portuguese Examples

1. A sobremesa de hoje é pudim. (Today's dessert is pudding.)
2. Eu adoro sobremesa de chocolate. (I love chocolate dessert.)
3. Não sobrou nenhuma sobremesa. (There was no dessert left.)
4. Qual é a sua sobremesa favorita? (What is your favorite dessert?)
5. Vamos pedir uma sobremesa? (Shall we order a dessert?)

Portuguese (Portugal) Examples

1. A sobremesa de hoje é pudim. (Today's dessert is pudding.)
2. Eu adoro sobremesa de chocolate. (I love chocolate dessert.)
3. Não sobrou nenhuma sobremesa. (There was no dessert left.)
4. Qual é a sua sobremesa preferida? (What is your preferred dessert?)
5. Gostaria de pedir uma sobremesa? (Would you like to order a dessert?)