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

Yes

The word "morto" is identical in meaning, grammar, and spelling in both Brazilian and European Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the vowels are more open and the final "o" is clearly pronounced as [u]. In Continental (European) Portuguese, unstressed vowels are often "reduced" or "swallowed," making the final "o" much shorter, subtler, and more neutralized.

Brazilian Portuguese

1. O meu cachorro está morto. (My dog is dead.)
2. Eu estou morto de cansaço. (I am dead tired.)
3. A planta está morta. (The plant is dead.)
4. Ele ficou morto de medo. (He was scared to death.)
5. A bateria do celular está morta. (The cell phone battery is dead.)

European Portuguese

1. O meu cão está morto. (My dog is dead.) Note: "Cão" is the natural term in Portugal.
2. Estou morto de cansaço. (I am dead tired.)
3. A planta está morta. (The plant is dead.)
4. Ele ficou morto de medo. (He was scared to death.)
5. O telemóvel ficou sem bateria. (The mobile phone is out of battery.) Note: "Telemóvel" is the natural term in Portugal, and "sem bateria" is the more natural way to express a dead battery.