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

No

While the fundamental meaning of "seguir" (to follow, to continue, or to proceed) is the same in both dialects, the answer is No because there are significant differences in typical usage and grammar (syntax).

In Brazilian Portuguese, "seguir" is commonly used with the gerund (e.g., seguindo) to indicate an ongoing action. In Continental Portuguese, the standard construction is the preposition "a" followed by the infinitive (e.g., a seguir). Additionally, there is a difference in how pronouns are used with the verb: Brazilians frequently use proclisis (placing the pronoun before the verb, e.g., me segue), whereas Portuguese speakers naturally use enclisis (placing the pronoun after the verb, e.g., segue-me).

Brazilian Portuguese Usage

1. Ele segue trabalhando muito. (He keeps working a lot.)
2. Siga as instruções no manual. (Follow the instructions in the manual.)
3. Eu sigo estudando todos os dias. (I keep studying every day.)
4. Ela está seguindo o caminho certo. (She is following the right path.)
5. Me segue no Instagram. (Follow me on Instagram.)

Continental Portuguese Usage

1. Ele segue a trabalhar muito. (He keeps working a lot.)
2. Segue as instruções no manual. (Follow the instructions in the manual.)
3. Eu sigo a estudar todos os dias. (I keep studying every day.)
4. Ela está a seguir o caminho certo. (She is following the right path.)
5. Segue-me no Instagram. (Follow me on Instagram.)