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

No

The difference lies in typical usage. In Brazil, the word "videogame" is the universal term used to refer to both the software (the games themselves) and the hardware (the console/machine). In Portugal, "videogame" is considered an anglicism and is much less common. Instead, Portuguese people naturally use the word videojogo to refer to the games and consola to refer to the hardware/machine.

Brazilian Portuguese usage:

1. Eu adoro jogar videogame. (I love playing video games.)
2. Meu videogame quebrou ontem. (My video game console broke yesterday.)
3. Ele passa o dia inteiro no videogame. (He spends the whole day on the video game.)
4. Comprei um videogame novo. (I bought a new video game console.)
5. Você quer jogar videogame comigo? (Do you want to play video games with me?)

European Portuguese usage:

1. Eu adoro jogar videojogos. (I love playing video games.)
2. A minha consola avariou ontem. (My console broke yesterday.)
3. Ele passa o dia inteiro nos videojogos. (He spends the whole day on video games.)
4. Comprei uma consola nova. (I bought a new console.)
5. Queres jogar um videojogo comigo? (Do you want to play a video game with me?)