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

No

While the core dictionary definition of "bater" (to hit, to knock, to strike, or to beat) is the same in both dialects, there are differences in typical usage and grammar (prepositions). In Brazil, it is common to use the preposition "na" (e.g., bater na porta), whereas in Portugal, the preposition "à" is the standard (e.g., bater à porta). Additionally, certain idiomatic expressions using "bater" are highly characteristic of Brazilian Portuguese, whereas a person from Portugal would naturally use different verbs to express the same concept.

Brazilian Portuguese usage

1. Eu vou bater na sua porta mais tarde. (I am going to knock on your door later.)
2. Ele bateu o carro na avenida principal. (He crashed the car on the main avenue.)
3. Vamos bater um papo depois do trabalho? (Shall we have a chat after work?)
4. Você precisa bater bem a massa do bolo. (You need to beat the cake batter well.)
5. Bateu uma vontade de comer chocolate. (A craving for chocolate hit me / I suddenly felt like eating chocolate.)

Continental Portuguese usage

1. Eu vou bater à tua porta mais tarde. (I am going to knock at your door later.)
2. Ele bateu com o carro na avenida principal. (He crashed the car on the main avenue.)
3. Vamos conversar um pouco depois do trabalho? (Shall we talk a bit after work?)
4. É preciso bater bem a massa do bolo. (It is necessary to beat the cake batter well.)
5. Deu-me uma vontade de comer chocolate. (I got a craving for chocolate.)