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

Yes

The spelling, meaning, and grammatical function of "tosse" (the noun for "cough") are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels tend to be more open and the final "e" is clearly pronounced as an [i] sound. In Continental Portuguese, the vowels are more closed, and the final "e" is often reduced to a nearly silent, very short sound [ɨ].

Brazilian Portuguese Examples

1. Eu estou com uma tosse muito forte. (I have a very strong cough.)
2. A tosse dele parece que não passa. (His cough seems like it won't go away.)
3. Você tomou algum remédio para a tosse? (Did you take any medicine for the cough?)
4. Essa tosse seca está me incomodando. (This dry cough is bothering me.)
5. A criança começou com uma tosse de madrugada. (The child started with a cough in the middle of the night.)

Continental Portuguese Examples

1. Estou com uma tosse terrível. (I have a terrible cough.)
2. A tosse da miúda está muito má. (The little girl's cough is very bad.)
3. Precisas de algum xarope para a tosse? (Do you need any syrup for the cough?)
4. Ele tem uma tosse com expetoração. (He has a cough with phlegm.)
5. Esta tosse é bastante irritante. (This cough is quite irritating.)