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

No

The word "trem" is not the same in both dialects due to differences in vocabulary and regional usage.

In Brazilian Portuguese, "trem" is the standard word for a train (railway vehicle). Additionally, in the Brazilian state of Minas Gerais, "trem" is used extensively as a colloquialism to mean "thing," "object," or "situation."

In Continental Portuguese, the word for train is comboio. A Portuguese person would not use "trem" to refer to a railway vehicle. Furthermore, the usage of "trem" to mean "thing" does not exist in Portugal; a Portuguese person would use coisa or objeto.

Brazilian Portuguese Examples

1. O trem acabou de passar pela estação. (The train just passed through the station.)
2. Eu costumo viajar de trem para visitar minha família. (I usually travel by train to visit my family.)
3. O que é esse trem aí na sua mão? (What is that thing in your hand? — Minas Gerais regionalism)
4. Me passa esse trem aqui, por favor. (Pass me this thing here, please. — Minas Gerais regionalism)
5. O trem de carga é muito barulhento. (The freight train is very noisy.)

Continental Portuguese Examples

1. O comboio acabou de passar pela estação. (The train just passed through the station.)
2. Eu costumo viajar de comboio para visitar a minha família. (I usually travel by train to visit my family.)
3. O que é essa coisa aí na tua mão? (What is that thing in your hand?)
4. Dá-me esse objeto aqui, por favor. (Give me this object here, please.)
5. O comboio de mercadorias é muito barulhento. (The freight train is very noisy.)