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

No

While the fundamental meanings of "carta" (mail, playing cards, or documents) are identical in both dialects, there is a difference in typical usage regarding restaurant menus. In Brazil, "carta" is used for a wine list (carta de vinhos), but for a food menu, the word "cardápio" is the standard. In Portugal, "a carta" or "a ementa" is frequently used to refer to the food menu itself. Additionally, there is a difference in pronunciation: Brazilian Portuguese features more open vowels, whereas Continental Portuguese features more closed or reduced vowels.

Brazilian Portuguese Examples

• Eu escrevi uma carta para você. (I wrote a letter to you.)
• Vamos jogar uma partida de cartas? (Shall we play a game of cards?)
• O garçom trouxe a carta de vinhos. (The waiter brought the wine list.)
• Ele assinou a carta de intenções. (He signed the letter of intent.)
• A carta náutica é muito importante. (The nautical chart is very important.)

Portuguese (Continental) Examples

• Eu escrevi uma carta para ti. (I wrote a letter to you.)
• Vamos jogar uma partida de cartas? (Shall we play a game of cards?)
• O empregado trouxe a ementa. (The waiter brought the menu.)
• Ele assinou a carta de intenções. (He signed the letter of intent.)
• A carta náutica é muito importante. (The nautical chart is very important.)