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

Yes

The word "idade" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the unstressed final "e" is typically pronounced as a clear [i] (as in the English word "see"). In Continental Portuguese, unstressed vowels are often reduced or "swallowed," so the final "e" sounds more like a closed [ɨ] (a muffled sound that is halfway between an "e" and an "i").

Brazilian Portuguese Examples

1. Qual é a sua idade? (What is your age?)
2. A idade não importa para o amor. (Age does not matter for love.)
3. Ele tem uma idade avançada. (He is of an advanced age.)
4. Nessa idade, as crianças são muito curiosas. (At that age, children are very curious.)
5. Ela está chegando a uma idade difícil. (She is reaching a difficult age.)

Continental Portuguese Examples

1. Quantos anos tens? (How old are you?)
2. A idade não conta no amor. (Age does not count in love.)
3. Ele tem uma idade avançada. (He is of an advanced age.)
4. Nessa idade, as crianças são muito curiosas. (At that age, children are very curious.)
5. Ela está a chegar a uma idade difícil. (She is reaching a difficult age.)