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

No

While the spelling and core meaning of "digo" (the first-person singular present of the verb dizer) are identical, there are significant differences in its grammar and typical usage. Specifically, the placement of object pronouns (clitics) differs: Brazilian Portuguese frequently uses proclisis (placing the pronoun before the verb, e.g., "te digo"), whereas Continental Portuguese uses enclisis (placing the pronoun after the verb, e.g., "digo-te"). Additionally, the subject pronouns paired with "digo" in conversation differ, as Brazilians typically use "você" and Portuguese people use "tu" for informal address.

Brazilian Portuguese Examples

1. Eu te digo a verdade sempre. (I always tell you the truth.)
2. O que eu digo não importa. (What I say doesn't matter.)
3. Eu digo que você está certo. (I say that you are right.)
4. Eu digo... digo, eu quis dizer outra coisa. (I say... I mean, I meant something else.)
5. Como eu digo isso em português? (How do I say this in Portuguese?)

Continental Portuguese Examples

1. Eu digo-te a verdade sempre. (I always tell you the truth.)
2. O que eu digo não interessa. (What I say is of no interest.)
3. Eu digo que tu estás certo. (I say that you are right.)
4. Eu digo... digo, eu queria dizer outra coisa. (I say... I mean, I wanted to say something else.)
5. Como é que eu digo isto em português? (How is it that I say this in Portuguese?)