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

No.

While the spelling and meaning of the name "Ana" are identical in both dialects, the answer is No because there are significant differences in typical usage. In Continental Portuguese, it is grammatically standard to use a definite article ("a" Ana) when referring to a person in the third person. In Brazilian Portuguese, the use of the article is optional and frequently omitted. Additionally, the syntactic structures used in conjunction with the name (such as the use of the gerund in Brazil versus the "a + infinitive" construction in Portugal) and the choice of second-person pronouns differ.

Brazilian Portuguese Examples

1. Ana é minha amiga. (Ana is my friend.)
2. Ana está estudando. (Ana is studying.)
3. Você conhece Ana? (Do you know Ana?)
4. Ana foi ao shopping. (Ana went to the mall.)
5. Ana é muito legal. (Ana is very cool.)

Continental Portuguese Examples

1. A Ana é minha amiga. (Ana is my friend.)
2. A Ana está a estudar. (Ana is studying.)
3. Tu conheces a Ana? (Do you know Ana?)
4. A Ana foi ao centro comercial. (Ana went to the shopping center.)
5. A Ana é muito simpática. (Ana is very nice.)