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

Yes

The spelling, meaning, and grammatical function of "honestamente" are identical in both Brazilian and European Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are more open and the rhythm is more "syllable-timed," meaning each syllable is pronounced clearly. In European Portuguese, the pronunciation is more "stress-timed," and unstressed vowels (such as the "e" in the suffix -mente) are often reduced or almost entirely silent, making the word sound much shorter and more "clipped" to a Brazilian ear.

Brazilian Portuguese Examples

1. Honestamente, eu não sei o que dizer. (Honestly, I don't know what to say.)
2. Acho que, honestamente, ele não vem hoje. (I think that, honestly, he isn't coming today.)
3. Honestamente, eu não concordo com você. (Honestly, I don't agree with you.)
4. Eu, honestamente, não queria ter ido à festa. (I, honestly, didn't want to have gone to the party.)
5. Honestamente, isso é muito complicado de explicar. (Honestly, this is very complicated to explain.)

European Portuguese Examples

While "honestamente" is used in Portugal, a Portuguese person would more naturally use "sinceramente" or "para ser sincero" to express the same sentiment.

1. Sinceramente, não faço ideia do que dizer. (Sincerely, I have no idea what to say.)
2. Para ser sincero, acho que ele não vem hoje. (To be sincere, I think he isn't coming today.)
3. Sinceramente, não concordo contigo. (Sincerely, I don't agree with you.)
4. Eu, para ser sincero, não queria ter ido à festa. (I, to be sincere, didn't want to have gone to the party.)
5. Sinceramente, isto é muito difícil de explicar. (Sincerely, this is very difficult to explain.)