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

No

While the spelling, grammar, and fundamental dictionary meaning of "caminha" are identical in both Brazilian and Continental Portuguese, there is a difference in typical usage regarding the noun form. In Brazil, "caminha" is the standard, most common term used to refer to a pet's bed. In Portugal, while the word is perfectly understood, it is much more natural and frequent to use the word "cama" (bed) even when referring to an animal's resting place. Additionally, there is a difference in pronunciation: Brazilian Portuguese typically features more open vowels, whereas Continental Portuguese uses more closed, reduced vowels.

Brazilian Portuguese (using the word "caminha" as is natural):

1. O cachorro está dormindo na caminha. (The dog is sleeping in his pet bed.)
2. Comprei uma caminha nova para o meu gato. (I bought a new pet bed for my cat.)
3. A caminha do Totó está muito confortável. (Totó's pet bed is very comfortable.)
4. Preciso lavar a caminha do cachorro hoje. (I need to wash the dog's pet bed today.)
5. Coloque o cobertor sobre a caminha. (Put the blanket on the pet bed.)

Continental Portuguese (using "cama" as is natural):

1. O cão está a dormir na cama. (The dog is sleeping in his bed.)
2. Comprei uma cama nova para o meu gato. (I bought a new bed for my cat.)
3. A cama do Totó está muito confortável. (Totó's bed is very comfortable.)
4. Preciso lavar a cama do cão hoje. (I need to wash the dog's bed today.)
5. Coloque a manta sobre a cama. (Put the blanket on the bed.)