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

Yes.

The meaning, grammar, and spelling of "algum" are identical in both Brazilian and European Portuguese. It functions as an indefinite adjective or pronoun meaning "some" or "any." The only differences are found in pronunciation: Brazilian Portuguese tends to pronounce vowels more openly and clearly, whereas European Portuguese often reduces or "swallows" unstressed vowels, making the nasalization of the final "m" sound slightly different to the ear.

Brazilian Portuguese Examples

1. Você tem algum irmão? (Do you have any brothers?)
2. Eu preciso de algum tempo para pensar. (I need some time to think.)
3. Algum dia eu vou viajar para o Japão. (Some day I am going to travel to Japan.)
4. Você viu algum filme legal ontem? (Did you see any cool movie yesterday?)
5. Tem algum problema com o meu carro? (Is there any problem with my car?)

European Portuguese Examples

1. Tens algum irmão? (Do you have any brothers?)
2. Preciso de algum tempo para pensar. (I need some time to think.)
3. Algum dia irei viajar para o Japão. (Some day I will travel to Japan.)
4. Viste algum filme engraçado ontem? (Did you see any funny movie yesterday?)
5. Há algum problema com o meu carro? (Is there any problem with my car?)