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

Yes

The word "colher" is identical in both Brazilian and Continental Portuguese in terms of meaning, spelling, and grammar. It functions as both a noun (meaning "spoon") and a verb (meaning "to harvest," "to pick," or "to gather"). The only significant difference is pronunciation. In Brazilian Portuguese, the stressed "e" is typically pronounced more openly and clearly. In Continental Portuguese, there is a much stronger tendency toward vowel reduction, which often makes the unstressed or even the stressed vowels in certain phonetic environments sound more closed or "swallowed."

Brazilian Portuguese

1. Eu estou usando uma colher de sopa. (I am using a tablespoon.)
2. Você pode me passar a colher, por favor? (Can you pass me the spoon, please?)
3. Ele colheu todas as frutas hoje. (He harvested all the fruit today.)
4. Coloque uma colher de açúcar no café. (Put a spoonful of sugar in the coffee.)
5. A colher caiu no chão. (The spoon fell on the floor.)

Continental Portuguese

1. Eu estou a usar uma colher de sopa. (I am using a tablespoon.)
2. Podes passar-me a colher, por favor? (Can you pass me the spoon, please?)
3. Ele colheu todas as frutas hoje. (He harvested all the fruit today.)
4. Põe uma colher de açúcar no café. (Put a spoonful of sugar in the coffee.)
5. A colher caiu ao chão. (The spoon fell to the floor.)