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

No.

While the core meanings of "deixar"—to leave, to let/allow, or to abandon—are identical in both dialects, the answer is No because there are significant differences in grammar (specifically how the word interacts with object pronouns), typical usage (the vocabulary that surrounds the word), and pronunciation.

Grammar & Usage: In Brazilian Portuguese, especially in spoken language, it is common to use subject pronouns (like eu) after "delar" (e.g., deixa eu), whereas in Portugal, the object pronoun is grammatically required (deixa-me). Additionally, the words used alongside "deixar" often differ (e.g., celular vs. telemóvel).

Pronunciation: Brazilian Portuguese features more open, melodic vowels. Continental Portuguese is much more "consonantal" and features significant vowel reduction, making unstressed syllables sound much shorter or almost silent.

Brazilian Portuguese Examples:

1. Deixa eu te ajudar. (Let me help you.)
2. Eu esqueci de deixar meu celular na mesa. (I forgot to leave my cell phone on the table.)
3. Deixa de bobeira! (Stop being silly!)
4. Eu deixei minha mochila no ônibus. (I left my backpack on the bus.)
5. Deixa isso pra lá. (Let it go / Forget about it.)

European Portuguese Examples:

1. Deixa-me ajudar-te. (Let me help you.)
2. Esqueci-me de deixar o meu telemóvel na mesa. (I forgot to leave my mobile phone on the table.)
3. Deixa de parvoíce! (Stop being silly!)
4. Deixei a minha mochila no autocarro. (I left my backpack on the bus.)
5. Deixa isso por isso mesmo. (Let it go / Leave it at that.)