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

Yes

The word "nesta" is a contraction of the preposition em (in/on/at) and the demonstrative pronoun esta (this). There are no differences in meaning, grammar, or spelling between Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazil, the vowels are typically more open and clearly articulated. In Portugal, unstressed vowels are often reduced or "swallowed," meaning the "e" in "nesta" may be much shorter or almost silent, making it sound similar to "n'sta."

Brazilian Portuguese Examples

1. Nesta semana eu vou viajar de trem. (This week I am going to travel by train.)
2. Você mora nesta casa? (Do you live in this house?)
3. Nesta cidade faz muito calor. (In this city it is very hot.)
4. Coloquei meu celular nesta bolsa. (I put my cell phone in this bag.)
5. Nesta situação, eu não sei o que fazer. (In this situation, I don't know what to do.)

Continental Portuguese Examples

1. Nesta semana vou viajar de comboio. (This week I am going to travel by train.)
2. Tu vives nesta casa? (Do you live in this house?)
3. Nesta cidade o tempo está ótimo. (In this city the weather is great.)
4. Pus o meu telemóvel nesta mala. (I put my mobile phone in this bag.)
5. Nesta situação, não há nada a fazer. (In this situation, there is nothing to be done.)