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

Yes

The word "neto" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only differences are in pronunciation (the phonetic realization of the accent) and a subtle difference in natural usage regarding the use of the definite article before possessive pronouns. In Brazil, it is very common to omit the article (e.g., "meu neto"), whereas in Portugal, it is much more natural to include it (e.g., "o meu neto").

Brazilian Portuguese Examples

1. Meu neto adora comer sorvete. (My grandson loves to eat ice cream.)
2. Vou buscar meu neto na escola. (I am going to pick up my grandson from school.)
3. O meu neto é muito esperto. (My grandson is very smart.)
4. Trouxe um presente para o meu neto. (I brought a present for my grandson.)
5. Ele é um neto muito carinhoso. (He is a very affectionate grandson.)

Continental Portuguese Examples

1. O meu neto adora comer gelado. (My grandson loves to eat ice cream.)
2. Vou buscar o meu neto à escola. (I am going to pick up my grandson from school.)
3. O meu neto é muito esperto. (My grandson is very smart.)
4. Trouxe um presente para o meu neto. (I brought a present for my grandson.)
5. Ele é um neto muito afetuoso. (He is a very affectionate grandson.)