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

Yes

The only difference between the use of "bege" in Brazilian Portuguese and Continental Portuguese is the pronunciation of the final unstressed "e."

• In Brazilian Portuguese, the final "e" is pronounced as a clear /i/ (similar to the "ee" in "see").
• In Continental Portuguese, the final "e" is a much more closed, unstressed vowel, often sounding nearly silent or neutralized to the ear.

There are no differences in meaning, spelling, or grammatical usage.

Brazilian Portuguese examples:

1. Eu comprei uma calça bege. (I bought beige pants.)
2. A parede da sala é bege. (The living room wall is beige.)
3. Essa cor bege combina com tudo. (This beige color goes with everything.)
4. Ele estava usando uma camisa bege. (He was wearing a beige shirt.)
5. O tapete bege está bem sujo. (The beige rug is very dirty.)

Continental Portuguese examples:

1. Comprei umas calças bege. (I bought some beige trousers.)
2. A parede da sala é bege. (The living room wall is beige.)
3. Esta cor bege combina com tudo. (This beige color goes with everything.)
4. Ele estava a usar uma camisa bege. (He was wearing a beige shirt.)
5. O tapete bege está muito sujo. (The beige rug is very dirty.)