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

Yes

The word "desta" (a contraction of the preposition de and the demonstrative pronoun esta) is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels—especially the final "a"—tend to be much more reduced, closed, or "clipped," making the word sound shorter and more neutralized.

Brazilian Portuguese Examples:

1. Desta vez eu vou conseguir! (This time I am going to succeed!)
2. O que você acha desta música? (What do you think of this song?)
3. Desta maneira você pode se machucar. (In this way you might hurt yourself.)
4. Eu não gosto desta cor de tinta. (I don't like this color of paint.)
5. Desta semana, eu estarei muito ocupado. (This week, I will be very busy.)

Continental Portuguese Examples:

1. Desta vez eu conseguirei! (This time I will succeed!)
2. O que achas desta música? (What do you think of this song?)
3. Desta forma, podes magoar-te. (In this way, you might hurt yourself.)
4. Não gosto desta cor de tinta. (I don't like this color of paint.)
5. Desta semana, estarei muito ocupado. (This week, I will be very busy.)