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

No

The difference is not in the meaning of the word, but in its grammar and usage.

In Brazilian Portuguese, the verb lembrar is very commonly used transitively, meaning it does not require a reflexive pronoun to function. A Brazilian will naturally say "Eu lembro o nome" (I remember the name). While Brazilians do use the reflexive form (lembrar-se), they also frequently use the construction lembrar de (e.g., "Eu lembro de você").

In Continental Portuguese, using the verb transitively to mean "to remember" is considered grammatically incorrect. A Portuguese person will almost exclusively use the pronominal/reflexive form (lembrar-se). To express the same idea, they must use the pronoun, such as "Eu lembro-me do nome" (I remember the name).

Brazilian Portuguese Examples

1. Você lembra o meu nome? (Do you remember my name?)
2. Isso me lembra muito o Brasil. (That reminds me a lot of Brazil.)
3. Eu não lembro de onde coloquei o celular. (I don't remember where I put the cell phone.)
4. Você lembra daquela viagem? (Do you remember that trip?)
5. Ele sempre lembra de me avisar. (He always remembers to warn me.)

Portuguese (Portugal) Examples

1. Tu lembras-te do meu nome? (Do you remember my name?)
2. Isto faz-me lembrar muito o Brasil. (This reminds me a lot of Brazil.)
3. Não me lembro de onde pus o telemóvel. (I don't remember where I put the mobile phone.)
4. Lembras-te daquela viagem? (Do you remember that trip?)
5. Ele lembra-se sempre de me avisar. (He always remembers to warn me.)