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

No.

While the fundamental meanings of "caixa" (box, container, or cashier) are shared by both variants, there are significant differences in typical usage and the words it is commonly paired with. In Brazil, "caixa eletrônico" is the standard term for an ATM, whereas in Portugal, the term "multibanco" is almost exclusively used. Additionally, when referring to "cardboard," Brazilians use "papelão" while Portuguese people use "cartão." There are also differences in the adjectives used to describe things related to the word, such as how a line (fila) is described.

Brazilian Portuguese Examples

1. Preciso ir ao caixa eletrônico sacar dinheiro. (I need to go to the ATM to withdraw money.)
2. Coloque os brinquedos nesta caixa de papelão. (Put the toys in this cardboard box.)
3. A fila do caixa está muito grande. (The checkout line is very large.)
4. O caixa do supermercado foi muito gentil. (The supermarket cashier was very kind.)
5. Eu comprei uma caixa de som nova. (I bought a new speaker.)

Continental Portuguese Examples

1. Preciso de ir ao multibanco levantar dinheiro. (I need to go to the ATM to withdraw money.)
2. Coloca os brinquedos nesta caixa de cartão. (Put the toys in this cardboard box.)
3. A fila da caixa está muito comprida. (The checkout line is very long.)
4. O funcionário da caixa foi muito gentil. (The cashier was very kind.)
5. Eu comprei uma coluna nova. (I bought a new speaker.)