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

Yes

The word "podem" is identical in meaning, spelling, and grammatical function in both Brazilian and European Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the unstressed "e" is generally more audible or slightly reduced. In Continental (European) Portuguese, there is a much stronger tendency to reduce or "swallow" unstressed vowels, making the "e" sound very short, almost silent, or more like a closed "i".

Brazilian Portuguese examples:

1. Eles podem vir à festa. (They can come to the party.)
2. Vocês podem me avisar? (Can you [plural] let me know?)
3. Essas pessoas podem ser perigosas. (These people can be dangerous.)
4. Os alunos podem fazer o teste hoje. (The students can take the test today.)
5. Vocês podem dar uma olhada nisso? (Can you [plural] take a look at this?)

Portuguese (Portugal) examples:

1. Eles podem vir à festa. (They can come to the party.)
2. Podem avisar-me? (Can you [plural] let me know?)
3. Estas pessoas podem ser perigosas. (These people can be dangerous.)
4. Os alunos podem fazer o teste hoje. (The students can take the test today.)
5. Podem dar uma vista de olhos nisto? (Can you [plural] take a look at this?)