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

No

While the spelling, grammar, and fundamental dictionary meaning of "tranquilo" are identical in both dialects, there are differences in typical usage and pronunciation.

In Brazil, "tranquilo" is used extensively as a versatile, standalone interjection or slang to mean "no problem," "it's all good," or "don't worry." In Portugal, while the word is used as an adjective to describe something peaceful, it is much less common as a conversational filler or a synonym for "no problem." Additionally, the pronunciation differs: Brazilian Portuguese features more open vowels and a more musical, rhythmic cadence, whereas Continental Portuguese uses more closed, reduced, or "swallowed" vowels.

Brazilian Portuguese Usage

1. "Tranquilo, pode deixar comigo." (No problem, you can leave it to me.)
2. "Tudo tranquilo por aqui?" (Is everything chill/okay around here?)
3. "O dia hoje foi bem tranquilo." (The day today was very calm.)
4. "Ele é um cara muito tranquilo, não se estressa." (He is a very chill guy, he doesn't get stressed.)
5. "A gente se fala depois, tranquilo?" (We'll talk later, okay?)

Continental Portuguese Usage

1. "Não há problema, pode deixar comigo." (No problem, you can leave it to me.)
2. "Está tudo bem por aí?" (Is everything okay over there?)
3. "O dia hoje foi muito calmo." (The day today was very calm.)
4. "Ele é uma pessoa muito sossegada, não se estressa." (He is a very quiet/calm person, he doesn't get stressed.)
5. "Falamos depois, está bem?" (We'll talk later, okay?)