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

No

There is a difference in typical usage. While "planejar" is the standard and most common verb in Brazil, a person from Portugal would naturally use the verb planear to express the same idea. Additionally, there are differences in the grammar of continuous actions: Brazilians use the gerund (planejando), whereas Portuguese people use the preposition a followed by the infinitive (a planear).

Brazilian Portuguese Examples

1. Eu vou planejar minha viagem. (I am going to plan my trip.)
2. Nós estamos planejando a festa. (We are planning the party.)
3. Ele planejou o seu futuro. (He planned his future.)
4. Nós precisamos planejar o orçamento. (We need to plan the budget.)
5. Você planeja viajar amanhã? (Are you planning to travel tomorrow?)

Continental Portuguese Examples

1. Eu vou planear a minha viagem. (I am going to plan my trip.)
2. Nós estamos a planear a festa. (We are planning the party.)
3. Ele planeou o seu futuro. (He planned his future.)
4. Nós precisamos de planear o orçamento. (We need to plan the budget.)
5. Tu planeias viajar amanhã? (Are you planning to travel tomorrow?)