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

Yes

The word "desemprego" is identical in spelling, meaning, and grammar in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the vowels are more open and clearly articulated. In Continental Portuguese, unstressed vowels are often reduced or "closed," and the "s" in the middle of the word is frequently pronounced as a "sh" sound (palatalization), making the word sound more like dsh-em-PRE-gu.

Brazilian Portuguese Examples

1. O desemprego está crescendo no país. (Unemployment is growing in the country.)
2. A taxa de desemprego caiu este mês. (The unemployment rate fell this month.)
3. O desemprego é um grande problema para nós. (Unemployment is a big problem for us.)
4. Ela está sofrendo com o desemprego. (She is suffering from unemployment.)
5. O governo quer baixar o desemprego. (The government wants to lower unemployment.)

Portuguese (Portugal) Examples

1. O desemprego tem aumentado muito. (Unemployment has been increasing a lot.)
2. As taxas de desemprego estão a subir. (The unemployment rates are rising.)
3. O desemprego desceu ligeiramente. (Unemployment went down slightly.)
4. O desemprego é uma preocupação grave. (Unemployment is a serious concern.)
5. É preciso combater o desemprego. (It is necessary to fight unemployment.)