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

No

There is a difference in typical usage. In Brazil, videogame (singular) or videogames (plural) is the standard, natural term used in all contexts. In Portugal, while the English-influenced "videogames" is understood, the word videojogos (or the singular videojogo) is the standard, natural, and much more frequent term used in both casual and formal speech.

Brazilian Portuguese Examples:

1. Eu adoro jogar videogame. (I love playing video games.)
2. Os videogames são muito divertidos. (Video games are very fun.)
3. Ele gasta muito dinheiro com videogames. (He spends a lot of money on video games.)
4. Meu presente de aniversário foi um videogame. (My birthday present was a video game.)
5. Você prefere videogames de aventura ou de ação? (Do you prefer adventure or action video games?)

Continental Portuguese Examples:

1. Eu adoro jogar videojogos. (I love playing video games.)
2. Os videojogos são muito divertidos. (Video games are very fun.)
3. Ele gasta muito dinheiro em videojogos. (He spends a lot of money on video games.)
4. O meu presente de aniversário foi um videojogo. (My birthday present was a video game.)
5. Preferes videojogos de aventura ou de ação? (Do you prefer adventure or action video games?)