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

Yes

The word "Chile" in Portuguese refers specifically to the South American country. In both Brazilian and Continental Portuguese, the spelling, meaning, and grammatical function (as a proper noun) are identical. The only difference lies in pronunciation: in Brazilian Portuguese, the final unstressed "e" is typically pronounced as a clear [i] (as in "shee-lee"), whereas in Continental Portuguese, the final "e" is often more neutralized or reduced to a very short, closed vowel [ɨ].

Brazilian Portuguese Examples

1. Eu quero viajar para o Chile no próximo ano. (I want to travel to Chile next year.)
2. O Chile é um país muito longo. (Chile is a very long country.)
3. A economia do Chile está crescendo. (Chile's economy is growing.)
4. Eu gosto muito do Chile. (I like Chile very much.)
5. O Chile tem montanhas lindas. (Chile has beautiful mountains.)

Continental Portuguese Examples

1. Eu pretendo viajar ao Chile no próximo ano. (I intend to travel to Chile next year.)
2. O Chile é um país muito comprido. (Chile is a very long country.)
3. A economia do Chile está a crescer. (Chile's economy is growing.)
4. Eu gosto imenso do Chile. (I like Chile very much.)
5. O Chile tem montanhas magníficas. (Chile has magnificent mountains.)