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

No.

While "relaxado" in both dialects can mean "relaxed" (the opposite of tense), there is a significant difference in typical usage and connotation. In Brazilian Portuguese, "relaxado" is frequently used as a common slang term to describe someone who is negligent, careless, unprofessional, or lazy. It implies a lack of discipline or a "slacker" attitude. In Continental Portuguese, while the word exists, it is rarely used to criticize someone's work ethic or character in this way; instead, a Portuguese person would much more naturally use the word "desleixado" to convey the same idea of negligence or sloppiness.

Brazilian Portuguese Usage

1. Ele é muito relaxado com o trabalho. (He is very careless with work.)
2. Não seja tão relaxado, temos um compromisso. (Don't be so irresponsible, we have a commitment.)
3. O rapaz é meio relaxado com a aparência. (The guy is a bit sloppy with his appearance.)
4. Ele vive relaxado, não cumpre as regras. (He is always slacking off, he doesn't follow the rules.)
5. A organização dele é muito relaxada. (His organization is very negligent.)

Continental Portuguese Usage

1. Ele é muito desleixado com o trabalho. (He is very careless with work.)
2. Não sejas tão desleixado, temos um compromisso. (Don't be so careless, we have a commitment.)
3. O rapaz é um bocado desleixado com o aspeto. (The guy is a bit sloppy with his appearance.)
4. Ele é um desleixado, não cumpre as regras. (He is a slacker, he doesn't follow the rules.)
5. A organização dele é muito desleixada. (His organization is very negligent.)