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

No

In Brazil, a batido typically refers to a non-alcoholic blended drink made from fruit and milk or yogurt (similar to a milkshake or a smoothie). In Portugal, while the word can refer to fruit blends, the term is most iconically and characteristically used to describe an alcoholic beverage made by blending spirits (such as brandy or gin) with ice and often fruit or cream. Because the typical usage and primary connotation differ, the word is not "the same" in a practical sense.

Brazilian Portuguese Examples

1. Eu adoro um batido de morango no café da manhã. (I love a strawberry smoothie for breakfast.)
2. O batido de banana com aveia é muito nutritivo. (The banana and oat smoothie is very nutritious.)
3. Você quer um batido de chocolate bem gelado? (Do you want a very cold chocolate milkshake?)
4. Este batido de abacate é muito cremoso. (This avocado smoothie is very creamy.)
5. Ele sempre pede um batido de frutas tropicais. (He always orders a tropical fruit smoothie.)

Continental Portuguese Examples

1. Vou pedir um smoothie de morango. (I'm going to order a strawberry smoothie.)
2. O smoothie de banana é muito saudável. (The banana smoothie is very healthy.)
3. Queres um smoothie de chocolate bem fresco? (Do you want a very fresh chocolate smoothie?)
4. Este smoothie de abacate é uma delícia. (This avocado smoothie is delicious.)
5. Eu tomo um smoothie de frutas no pequeno-almoço. (I have a fruit smoothie for breakfast.)