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

Yes

The word "medo" is identical in meaning, spelling, and grammar across both Brazilian and Continental Portuguese. The only difference lies in pronunciation. Brazilian Portuguese tends to be more "syllable-timed" and musical, with more open and clearly articulated vowels. In contrast, Continental Portuguese is more "stress-timed" and can sound more "clipped" or "closed," as speakers often reduce or shorten unstressed vowels.

Brazilian Portuguese Examples

1. Eu tenho muito medo de escuro. (I am very afraid of the dark.)
2. Ele tá com muito medo da prova. (He is very afraid of the test.) Note: "tá" is a common natural contraction of "está" in Brazil.
3. Eu morro de medo de barata! (I am terrified of cockroaches!)
4. Não precisa ter medo de mim. (You don't need to be afraid of me.)
5. A gente tem muito medo de viajar de avião. (We are very afraid of traveling by plane.) Note: "A gente" is the most common natural way to say "we" in Brazil.

Continental Portuguese Examples

1. Tenho imenso medo de andar sozinho à noite. (I have immense fear of walking alone at night.) Note: "Imenso" is a very common way for a Portuguese person to add emphasis.
2. Ele ficou com um medo terrível. (He became terribly afraid.)
3. Não tenhas medo, está tudo bem. (Don't be afraid, everything is fine.) Note: "Tenhas" uses the second-person singular imperative/subjunctive typical of Portugal.
4. Sinto um certo medo de perder o meu emprego. (I feel a certain fear of losing my job.)
5. Ela estava com um medo enorme da tempestade. (She was feeling an enormous fear of the storm.)