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

Yes

The word "forno" is identical in both Brazilian and Continental Portuguese in terms of meaning, spelling, and grammar. The only difference lies in pronunciation. In Brazilian Portuguese, the final unstressed "o" is typically pronounced as a "u" sound (/fórnu/). In Continental Portuguese, the pronunciation of the stressed "o" is generally more closed, and while the final vowel is also reduced, the phonetic texture and rhythm of the word differ significantly due to the distinct accentuation patterns of Portugal.

Brazilian Portuguese

1. O forno já está quente. (The oven is already hot.)
2. Coloque o bolo no forno. (Put the cake in the oven.)
3. O forno está desligado. (The oven is turned off.)
4. Preciso comprar um forno novo. (I need to buy a new oven.)
5. Estou assando carne no forno. (I am roasting meat in the oven.)

Continental Portuguese

1. O forno já está quente. (The oven is already hot.)
2. Põe o bolo no forno. (Put the cake in the oven.)
3. O forno está desligado. (The oven is turned off.)
4. Preciso de comprar um forno novo. (I need to buy a new oven.)
5. Estou a assar carne no forno. (I am roasting meat in the oven.)