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

Yes

The word "bom" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the nasalized vowel is typically more open and resonant. In Continental Portuguese, the vowel tends to be more closed and the nasalization is often more subtle, sometimes sounding "swallowed" or almost muffled.

Brazilian Portuguese Examples

1. Tudo bom? (Everything good?)
2. O almoço estava muito bom. (Lunch was very good.)
3. Ele é um bom professor. (He is a good teacher.)
4. Bom dia, Maria! (Good morning, Maria!)
5. Que bom que você chegou. (How good that you arrived.)

Portuguese (Continental) Examples

1. Tudo bem? (Everything well? — In Portugal, "Tudo bem?" is the more natural/standard way to express the idea of "Tudo bom?")
2. O almoço estava muito bom. (Lunch was very good.)
3. Ele é um bom professor. (He is a good teacher.)
4. Bom dia, Maria! (Good morning, Maria!)
5. Que bom que vieste. (How good that you arrived. — A Portuguese person would naturally use the "tu" conjugation "vieste" instead of the Brazilian "você chegou").