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

No

While the spelling and the fundamental meaning of the word "dançar" are identical in both Brazilian and Continental Portuguese, the answer is No because there are significant differences in grammatical usage regarding continuous actions and differences in pronunciation.

Grammar and Usage: In Brazil, when expressing an ongoing action, the gerund is used (e.g., dançando). In Portugal, the standard construction is the preposition "a" followed by the infinitive (e.g., a dançar). Additionally, the choice of second-person pronouns (você in Brazil vs. tu in Portugal) changes how the verb is conjugated in natural conversation.

Pronunciation: Brazilian Portuguese features more open, melodic vowels and a more distinct, vibrant "r" at the end of words. Continental Portuguese features more "closed" or reduced vowels and a much more subtle, often guttural or nearly silent "r" at the end of verbs.

Brazilian Portuguese Examples:

1. Eu amo dançar samba. (I love dancing samba.)
2. Eles estão dançando muito bem. (They are dancing very well.)
3. Ela estava dançando na festa. (She was dancing at the party.)
4. Nós vamos dançar essa música. (We are going to dance to this song.)
5. Você quer dançar comigo? (Do you want to dance with me?)

Continental Portuguese Examples:

1. Eu adoro dançar salsa. (I love dancing salsa.)
2. Eles estão a dançar muito bem. (They are dancing very well.)
3. Ela estava a dançar na festa. (She was dancing at the party.)
4. Nós vamos dançar esta música. (We are going to dance to this song.)
5. Tu queres dançar comigo? (Do you want to dance with me?)