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

No

While the core meaning of "lembro" (the first-person singular present indicative of the verb lembrar) remains "I remember" in both dialects, there is a significant difference in usage and grammar.

In Brazilian Portuguese, it is very common and culturally natural to use the verb transitively (without a reflexive pronoun) or with the preposition "de" (e.g., "Eu lembro disso"). In Continental Portuguese, the verb is almost exclusively used in its pronominal/reflexive form (lembrar-se) followed by the preposition "de" (e.g., "Eu lembro-me disso"). Using "lembro" without the reflexive pronoun "me" sounds incomplete or ungrammatical to a Portuguese person.

Brazilian Portuguese Usage

Note: Brazilian usage often omits the reflexive pronoun "me" in casual speech.

1. Eu lembro do seu rosto. (I remember your face.)
2. Não lembro o que aconteceu ontem. (I don't remember what happened yesterday.)
3. Eu lembro muito bem daquela viagem. (I remember that trip very well.)
4. Você lembra de mim? (Do you remember me?)
5. Eu lembro que você me prometeu isso. (I remember that you promised me this.)

Continental Portuguese Usage

Note: Portuguese usage strictly employs the reflexive pronoun and follows standard placement rules.

1. Eu lembro-me do seu rosto. (I remember your face.)
2. Não me lembro do que aconteceu ontem. (I don't remember what happened yesterday.)
3. Eu lembro-me muito bem daquela viagem. (I remember that trip very well.)
4. Lembras-te de mim? (Do you remember me?)
5. Eu lembro-me que me prometeste isso. (I remember that you promised me this.)