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

No.

While the spelling and the literal meaning of the word are identical, there is a significant difference in typical usage and grammar regarding how continuous actions are expressed. In Brazil, the gerund (cuidando) is the standard way to express an action currently in progress. In Portugal, the construction a + infinitive (a cuidar) is used almost exclusively for this purpose. While a Portuguese person would understand "estou cuidando," it would sound distinctly Brazilian to them.

Brazilian Portuguese Examples

1. Estou cuidando do meu cachorro. (I am taking care of my dog.)
2. Ela está cuidando das plantas. (She is taking care of the plants.)
3. Nós estamos cuidando do projeto. (We are taking care of/managing the project.)
4. Ele está cuidando do irmão mais novo. (He is looking after his younger brother.)
5. Você está cuidando da sua saúde? (Are you taking care of your health?)

European Portuguese Examples

1. Estou a cuidar do meu cão. (I am taking care of my dog.)
2. Ela está a cuidar das plantas. (She is taking care of the plants.)
3. Nós estamos a cuidar do projeto. (We are taking care of/managing the project.)
4. Ele está a cuidar do irmão mais novo. (He is looking after his younger brother.)
5. Estás a cuidar da tua saúde? (Are you taking care of your health?)