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

No

The difference lies in the specificity of usage. In Brazilian Portuguese, the word "detergente" refers almost exclusively to liquid dish soap used for cleaning plates and cutlery. In Continental Portuguese, "detergente" is a much broader, generic term for any cleaning agent (including laundry detergent or floor cleaners). To refer specifically to dish soap, a person from Portugal will typically add a qualifier, such as "detergente da loiça" (dish detergent), or may use the word "lavaloiças."

Brazilian Portuguese Examples

1. Acabou o detergente. (The dish soap ran out.)
2. Passe um pouco de detergente na esponja. (Put a little dish soap on the sponge.)
3. Este detergente tem cheiro de limão. (This dish soap smells like lemon.)
4. Preciso comprar detergente e esponja. (I need to buy dish soap and a sponge.)
5. Use detergente para tirar a gordura da panela. (Use dish soap to remove the grease from the pan.)

Continental Portuguese Examples

1. Acabou o detergente da loiça. (The dish soap ran out.)
2. Preciso comprar detergente da roupa. (I need to buy laundry detergent.)
3. Este detergente da loiça é muito eficaz. (This dish soap is very effective.)
4. Use este detergente para limpar o chão. (Use this detergent to clean the floor.)
5. O detergente da roupa acabou. (The laundry detergent ran out.)