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

No

The difference lies in grammar and usage. While the core meaning of "lembrar" (to remember or to remind) remains the same, the way the verb is constructed differs significantly. In Brazilian Portuguese, it is very common in colloquial speech to use "lembrar" as a transitive direct verb, meaning the reflexive pronoun and the preposition "de" are often omitted (e.g., Eu lembrei o nome). In Continental Portuguese, the pronominal form "lembrar-se" is the standard and much more frequent way to express "to remember," and it almost always requires the preposition "de" (e.g., Lembrei-me do nome). Additionally, the placement of pronouns (clitics) differs, with Brazilians preferring them before the verb (me lembra) and Portuguese people preferring them after the verb (lembra-me).

Brazilian Portuguese Examples:

1. Eu lembrei o seu número. (I remembered your number.)
2. Não esquece de me lembrar do bolo. (Don't forget to remind me about the cake.)
3. Isso me lembra muito o Brasil. (This reminds me a lot of Brazil.)
4. Você lembra o que aconteceu ontem? (Do you remember what happened yesterday?)
5. Eu lembrei que esqueci a chave. (I remembered that I forgot the key.)

European Portuguese Examples:

1. Eu lembrei-me do teu número. (I remembered your number.)
2. Não te esqueças de me lembrar do bolo. (Don't forget to remind me about the cake.)
3. Isto lembra-me muito o Brasil. (This reminds me a lot of Brazil.)
4. Lembras-te do que aconteceu ontem? (Do you remember what happened yesterday?)
5. Lembrei-me de que esqueci a chave. (I remembered that I forgot the key.)