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

No

While the dictionary meaning and grammatical function of "para" are identical in both dialects, there are differences in typical usage, pronunciation, and informal spelling.

1. Usage: In Brazil, "para" is the standard preposition used to indicate movement or destination (e.g., "Vou para o Rio"). In Portugal, the preposition "a" (or its contractions ao/à) is much more naturally used for the same purpose (e.g., "Vou ao Rio").
2. Pronunciation and Spelling: In Brazilian Portuguese, "para" is almost universally reduced to "pra" in spoken conversation and is extremely common in informal writing. In Continental Portuguese, while "pra" exists, the full form "para" is much more strictly maintained in standard speech.

Brazilian Portuguese Examples

1. Eu vou para a escola amanhã. (I am going to school tomorrow.)
2. Este presente é para você. (This gift is for you.)
3. Eu estudo muito para passar na prova. (I study a lot to pass the exam.)
4. O ônibus para o centro está atrasado. (The bus to the center is late.)
5. Fiz este bolo para a sua festa. (I made this cake for your party.)

Continental Portuguese Examples

1. Eu vou à escola amanhã. (I am going to school tomorrow.)
2. Este presente é para ti. (This gift is for you.)
3. Eu estudo muito para passar no exame. (I study a lot to pass the exam.)
4. O autocarro para o centro está atrasado. (The bus to the center is late.)
5. Fiz este bolo para a tua festa. (I made this cake for your party.)