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

Yes.

The word "guardanapo" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference lies in pronunciation: in Brazilian Portuguese, the final unstressed "o" is typically pronounced as a "u" ([ɡwaɾda'napu]), whereas in Continental Portuguese, the vowel is more closed and the rhythmic stress pattern of the word differs slightly.

Brazilian Portuguese Examples

1. Você pode me passar um guardanapo? (Can you pass me a napkin?)
2. Eu usei o guardanapo para limpar a boca. (I used the napkin to clean my mouth.)
3. O restaurante esqueceu de colocar os guardanapos na mesa. (The restaurant forgot to put the napkins on the table.)
4. Pegue um guardanapo de papel para a criança, por favor. (Get a paper napkin for the child, please.)
5. O guardanapo caiu no chão. (The napkin fell on the floor.)

Continental Portuguese Examples

1. Podes passar-me um guardanapo? (Can you pass me a napkin?)
2. Usei o guardanapo para limpar a boca. (I used the napkin to clean my mouth.)
3. O restaurante esqueceu-se de colocar os guardanapos na mesa. (The restaurant forgot to put the napkins on the table.)
4. Pega um guardanapo de papel para a criança, por favor. (Get a paper napkin for the child, please.)
5. O guardanapo caiu ao chão. (The napkin fell to the floor.)