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

Yes

The meaning, grammar, and spelling of "desde" are identical in both Brazilian and Continental Portuguese. There are no differences in how the word functions or what it signifies. The only difference is pronunciation. In Brazilian Portuguese, vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels (like the "e" in "desde") tend to be more reduced, closed, or even "swallowed," making the word sound much shorter or more clipped to a non-native ear.

Brazilian Portuguese Examples

1. Eu trabalho aqui desde o ano passado. (I have worked here since last year.)
2. Ele não fala comigo desde ontem. (He hasn't spoken to me since yesterday.)
3. Desde que eu cheguei, estou muito feliz. (Since I arrived, I have been very happy.)
4. Estou esperando você desde as oito horas. (I have been waiting for you since eight o'clock.)
5. A loja fica aberta desde cedo. (The store has been open since early.)

Portuguese (Portugal) Examples

1. Eu trabalho aqui desde o ano passado. (I have worked here since last year.)
2. Ele não me fala desde ontem. (He hasn't spoken to me since yesterday.)
3. Desde que tu chegaste, estou muito feliz. (Since you arrived, I have been very happy.)
4. Estou à espera de ti desde as oito horas. (I have been waiting for you since eight o'clock.)
5. A loja está aberta desde cedo. (The store has been open since early.)