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

No

The literal meaning of "pepino" is identical in both dialects (the vegetable). However, there is a difference in typical usage. In Brazil, "peperino" is frequently used as a slang term to describe a "problem," "mess," or "difficult situation" (e.g., se meter num pepino). In Portugal, the word is almost exclusively used to refer to the vegetable itself. There is also a difference in pronunciation; Brazilians tend to pronounce the final unstressed "o" as a clear "u" sound, while in Portugal, the vowel is more closed and reduced.

Brazilian Portuguese Examples

1. Eu adoro comer salada com pepino. (I love eating salad with cucumber.)
2. Comprei dois pepinos na feira hoje. (I bought two cucumbers at the market today.)
3. Nossa, eu me meti em um pepino enorme! (Wow, I got myself into a huge mess!)
4. O sanduíche de pepino é muito refrescante. (The cucumber sandwich is very refreshing.)
5. Ele sempre arruma um pepino para resolver. (He always finds a problem to solve.)

Continental Portuguese Examples

1. Eu adoro comer salada com pepino. (I love eating salad with cucumber.)
2. Comprei dois pepinos no supermercado. (I bought two cucumbers at the supermarket.)
3. Caramba, eu meti-me num sarilho enorme! (Gosh, I got myself into a huge mess!)
4. A sanduíche de pepino é muito refrescante. (The cucumber sandwich is very refreshing.)
5. Ele está sempre a arranjar um problema para resolver. (He is always finding a problem to solve.)