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

Yes

The word "perder" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The differences between the two varieties lie in pronunciation (specifically vowel openness and rhythm) and the surrounding vocabulary or syntactic structures used in a sentence. While the verb remains the same, a Brazilian might use a gerund ("perdendo") whereas a Portuguese person would use the prepositional infinitive ("a perder"), and the objects being lost (like "bus" or "cellphone") have different regional names.

Brazilian Portuguese Examples:

1. Eu perdi minha chave. (I lost my key.)
2. Eu perdi o jogo. (I lost the game.)
3. Ele está perdendo o ônibus. (He is missing the bus.)
4. Eu perdi meu celular. (I lost my cell phone.)
5. Nós perdemos a chance. (We lost the chance.)

Portuguese (Continental) Examples:

1. Eu perdi a minha chave. (I lost my key.)
2. Eu perdi o jogo. (I lost the game.)
3. Ele está a perder o autocarro. (He is missing the bus.)
4. Eu perdi o meu telemóvel. (I lost my mobile phone.)
5. Nós perdemos a oportunidade. (We lost the opportunity.)