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

No

While the spelling and the fundamental meaning of "esqueci" (the first-person singular past tense of esquecer) are identical in both dialects, there is a significant difference in typical usage regarding grammar.

In Brazilian Portuguese, it is very common and natural to use the verb in its non-pronominal form (esqueci algo) or to use the preposition "de" (esqueci de algo). In Continental Portuguese, the reflexive/pronominal form (esqueci-me de algo) is the standard and most natural way to express the idea of forgetting something. Additionally, certain vocabulary related to the context of forgetting (like "cell phone" or "bus") differs between the two regions.

Brazilian Portuguese Examples

1. Esqueci meu celular em casa. (I forgot my cell phone at home.)
2. Esqueci de pegar a chave. (I forgot to grab the key.)
3. Esqueci o nome dele. (I forgot his name.)
4. Esqueci de te contar a novidade. (I forgot to tell you the news.)
5. Esqueci a mochila no ônibus. (I forgot the backpack on the bus.)

Continental Portuguese Examples

1. Esqueci-me do telemóvel em casa. (I forgot my mobile phone at home.)
2. Esqueci-me de apanhar a chave. (I forgot to grab the key.)
3. Esqueci-me do nome dele. (I forget his name.)
4. Esqueci-me de te dizer a novidade. (I forgot to tell you the news.)
5. Esqueci-me da mochila no autocarro. (I forgot the backpack on the bus.)