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

No

While the primary meaning of "batata" is identical in both dialects (the vegetable known as potato), there are differences in typical usage and idiomatic meaning. In Brazil, "batata" can be used as slang to indicate certainty or a "sure thing" (e.g., é batata). In Portugal, the word is almost exclusively used to refer to the vegetable. Additionally, there is a difference in how the word is used in the plural for certain dishes; Brazilians typically use the singular/collective "batata frita" for french fries, whereas Portuguese people more naturally use the plural "batatas fritas."

Brazilian Portuguese Examples:

1. Eu adoro comer batata frita com ketchup. (I love eating french fries with ketchup.)
2. O purê de batata ficou muito cremoso. (The mashed potato turned out very creamy.)
3. Eu prefiro batata doce assada. (I prefer roasted sweet potato.)
4. Se você estudar muito, é batata que você passa na prova. (If you study hard, it's a sure thing you'll pass the exam.)
5. Sobrou um pouco de batata no prato. (There is a little potato left on the plate.)

Portuguese (Portugal) Examples:

1. Eu adoro comer batatas fritas com ketchup. (I love eating french fries with ketchup.)
2. O puré de batata ficou muito cremoso. (The mashed potato turned out very creamy.)
3. Eu prefiro batata doce assada. (I prefer roasted sweet potato.)
4. Se estudares muito, é certo que passas no exame. (If you study hard, it is certain that you will pass the exam.)
5. Sobrou um pouco de batata no prato. (There is a little potato left on the plate.)