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

Yes

The spelling, meaning, and grammatical category of the word "táxi" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the final unstressed "i" is typically more open and clearly articulated. In Continental Portuguese, unstressed vowels like the final "i" are much more closed and often reduced or neutralized in rapid speech.

Brazilian Portuguese Examples:

1. Vou pegar um táxi para ir ao aeroporto. (I'm going to take a taxi to go to the airport.)
2. O táxi demorou muito para chegar. (The taxi took a long time to arrive.)
3. Você pode chamar um táxi para mim? (Can you call a taxi for me?)
4. O preço do táxi estava muito caro hoje. (The taxi price was very expensive today.)
5. Deixei minha bolsa dentro do táxi. (I left my bag/purse inside the taxi.)

Continental Portuguese Examples:

1. Vou apanhar um táxi para ir ao aeroporto. (I'm going to take a taxi to go to the airport.)
2. O táxi demorou imenso tempo a chegar. (The taxi took a very long time to arrive.)
3. Pode chamar-me um táxi, por favor? (Can you call a taxi for me, please?)
4. O preço do táxi estava caríssimo hoje. (The taxi price was very expensive today.)
5. Esqueci a minha mala dentro do táxi. (I forgot my bag/suitcase inside the taxi.)