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

Yes

The word "plano" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation and the rhythm of the surrounding sentence. Brazilian Portuguese tends to have more open, melodic vowels and a more "syllable-timed" rhythm. In contrast, Continental Portuguese features significant vowel reduction, where unstressed vowels are shortened or almost "swallowed," giving the language a more "stress-timed" and clipped sound.

Brazilian Portuguese Examples

1. Eu tenho um plano para o final de semana. (I have a plan for the weekend.)
2. O terreno aqui é bem plano. (The terrain here is very flat.)
3. Qual é o seu plano para amanhã? (What is your plan for tomorrow?)
4. Precisamos de um plano de ação. (We need an action plan.)
5. Esse plano não vai dar certo. (This plan is not going to work.)

Continental Portuguese Examples

1. Tenho um plano para o fim de semana. (I have a plan for the weekend.)
2. O terreno aqui é bastante plano. (The terrain here is quite flat.)
3. Qual é o teu plano para amanhã? (What is your plan for tomorrow?)
4. Precisamos de um plano de ação. (We need an action plan.)
5. Este plano não vai resultar. (This plan is not going to work/succeed.)