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

Yes

The word "lei" (meaning law, rule, or statute) is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the diphthong "ei" is typically more open, and the final "i" is clearly and distinctly articulated. In Continental Portuguese, the pronunciation is more "closed" or reduced; the final "i" is often much more subtle or nearly swallowed, making the word sound more compressed.

Brazilian Portuguese Examples

1. A lei é igual para todos. (The law is the same for everyone.)
2. Ele não pode desobedecer a lei. (He cannot disobey the law.)
3. Essa lei é muito antiga. (This law is very old.)
4. A nova lei já está valendo. (The new law is already in effect.)
5. É lei que o aluno estude. (It is a rule that the student studies.)

Portuguese (Portugal) Examples

1. É proibido por lei fumar aqui. (It is forbidden by law to smoke here.)
2. A lei foi alterada recentemente. (The law was recently changed.)
3. Temos de respeitar a lei. (We must respect the law.)
4. A lei protege os direitos laborais. (The law protects labor rights.)
5. Segundo a lei, é obrigatório. (According to the law, it is mandatory.)