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

Yes

The word "diário" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. It functions as a noun (meaning "diary" or "logbook") and as an adjective (meaning "daily"). The only differences between the two variants are in pronunciation. In many Brazilian dialects, the "d" is palatalized (sounding like "dj"), and the "r" may be a retroflex [ɻ] in certain regions. In Continental Portuguese, the "d" remains a dental consonant [d], and the unstressed vowels are generally more closed and less reduced than in Brazilian Portuguese.

Brazilian Portuguese

1. Eu escrevo no meu diário todas as noites. (I write in my diary every night.)
2. Ele tem uma rotina diária muito cansativa. (He has a very tiring daily routine.)
3. O professor anotou a falta no diário de classe. (The teacher noted the absence in the class register.)
4. O preço diário do ônibus é cinco reais. (The daily price of the bus is five reais.)
5. O custo diário do hotel é muito alto. (The daily cost of the hotel is very high.)

Portuguese (Portugal)

1. Eu escrevo no meu diário todas as noites. (I write in my diary every night.)
2. Ele tem uma rotina diária muito cansativa. (He has a very tiring daily routine.)
3. O professor anotou a falta no diário de classe. (The teacher noted the absence in the class register.)
4. O preço diário do autocarro é cinco euros. (The daily price of the bus is five euros.)
5. O custo diário do hotel é muito alto. (The daily cost of the hotel is very high.)