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

No.

The difference lies in usage and pragmatics. In Brazilian Portuguese, "a gente" is the standard, colloquial way to express the first-person plural ("we"). While it is grammatically third-person singular, Brazilians use it almost interchangeably with "nós" in everyday conversation. In Continental Portuguese, while "a gente" can technically be used to mean "we," it is much less frequent and carries a significantly more informal or "commoner" connotation. A Portuguese person will naturally use the pronoun "nós" to express "we," reserving "a gente" for referring to "people" or "folks" in a general, collective sense.

Brazilian Portuguese Usage

1. A gente vai ao cinema amanhã. (We are going to the movies tomorrow.)
2. A gente precisa conversar sobre isso. (We need to talk about this.)
3. O que a gente faz agora? (What do we do now?)
4. A gente gosta muito de você. (We like you a lot.)
5. A gente se vê mais tarde. (We will see each other later.)

Continental Portuguese Usage

1. Nós vamos ao cinema amanhã. (We are going to the movies tomorrow.)
2. Nós precisamos conversar sobre isto. (We need to talk about this.)
3. O que é que nós fazemos agora? (What do we do now?)
4. Nós gostamos muito de ti. (We like you a lot.)
5. Vemo-nos mais tarde. (We will see each other later.)