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

No.

While the spelling and the fundamental dictionary meaning of "fico" (the first-person singular present of the verb ficar) are the same, the typical usage and the grammatical structures surrounding the word differ significantly between the two dialects.

In Brazilian Portuguese, "fico" is frequently used with the gerund (e.g., fico estudando) to describe ongoing actions. In Continental Portuguese, it is much more natural to use the preposition "a" followed by the infinitive (e.g., fico a estudar). Additionally, while the word "fico" is used in both to denote a change in state (becoming/getting), the vocabulary used to describe those states often differs (e.g., bravo in Brazil vs. zangado in Portugal), and the pronouns used in conjunction with the verb often change (você vs. tu).

Brazilian Portuguese Natural Usage:

1. Eu fico muito feliz quando você vem. (I get very happy when you come.)
2. Eu fico em casa no fim de semana. (I stay at home on the weekend.)
3. Eu fico estudando o dia todo. (I stay studying all day.)
4. Eu fico brava com isso. (I get angry with that.)
5. Eu fico aqui te esperando. (I'll stay here waiting for you.)

Continental Portuguese Natural Usage:

1. Eu fico muito feliz quando tu vens. (I get very happy when you come.)
2. Eu fico em casa ao fim de semana. (I stay at home on the weekend.)
3. Eu fico a estudar o dia todo. (I stay studying all day.)
4. Eu fico zangada com isso. (I get angry with that.)
5. Eu fico aqui à espera de ti. (I'll stay here waiting for you.)