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

No

While the core meaning, spelling, and grammatical category of the word "tudo" are identical in both dialects, the answer is No because of differences in typical usage and pronunciation.

In Brazil, "tudo" is frequently used in highly condensed, shorthand forms as a standalone greeting (e.g., "Tudo?" or "Tudo bom?"). In Portugal, while these are understood, speakers much more naturally prefer complete syntactic structures, almost always including the verb estar (e.g., "Está tudo bem?"). Additionally, the pronunciation differs significantly: Brazilian Portuguese features more open, melodic vowels, whereas European Portuguese is more "closed," often reducing or "swallowing" unstressed vowels, which gives the word a sharper, more clipped sound.

Brazilian Portuguese

1. Tudo bem com você? (Is everything well with you?)
2. Você comeu tudo sozinho? (Did you eat everything by yourself?)
3. Tudo certo por aqui. (Everything is alright here.)
4. Eu quero tudo o que tem lá! (I want everything that is there!)
5. Tudo ótimo, e você? (Everything is great, and you?)

European Portuguese

1. Está tudo bem contigo? (Is everything well with you?)
2. Comeste tudo? (Did you eat everything?)
3. Está tudo em ordem por cá. (Everything is in order around here.)
4. Eu quero tudo o que está lá! (I want everything that is there!)
5. Está tudo ótimo, e contigo? (Everything is great, and with you?)