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

No

While the core definition of the word remains the same, there are significant differences in pronunciation and typical usage.

1. Pronunciation: Brazilian Portuguese features more "open" and clearly articulated vowels. In Continental Portuguese, unstressed vowels undergo heavy reduction, often becoming "closed" or nearly silent, which can change the perceived rhythm of the word.
2. Typical Usage: In Brazil, "cuidar" is frequently used to mean "to handle," "to manage," or "to deal with" a task or problem. In Portugal, while "cuidar" is used for nurturing or maintenance, the expression "tratar de" is much more natural when expressing the idea of handling a task or resolving an issue.
3. Syntax: Although the verb itself is the same, the grammatical construction around it differs. Brazilians typically use the gerund (e.g., cuidando), whereas Portuguese people use the preposition "a" followed by the infinitive (e.g., a cuidar).

Brazilian Portuguese

• Eu cuido do meu cachorro. (I take care of my dog.)
• Pode deixar que eu cuido disso. (Don't worry, I'll handle that.)
• Ela está cuidando do jardim. (She is taking care of the garden.)
• Você precisa cuidar da sua saúde. (You need to take care of your health.)
• Ele cuida da própria vida. (He minds his own business.)

Continental Portuguese

• Eu cuido do meu cão. (I take care of my dog.)
• Podes deixar que eu trato disso. (Don't worry, I'll handle that.)
• Ela está a cuidar do jardim. (She is taking care of the garden.)
• Tens de cuidar da tua saúde. (You need to take care of your health.)
• Ele cuida da própria vida. (He minds his own business.)