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

No

While the fundamental meaning of "cobertor" is the same in both dialects (a thick, warm cloth used for bedding), there is a difference in typical usage and pronunciation:

• Usage: In Brazil, "cobertor" is the standard, generic term for any thick or heavy blanket. In Portugal, while "cobertor" is used to describe specifically heavy or thick blankets, the word "manta" is much more frequently used as the natural, general term for various types of bed covers, throws, or lighter blankets.
• Pronunciation: In Brazilian Portuguese, the "e" in the second syllable is typically more open. In Continental Portuguese, this "e" is much more closed or even reduced to a near-silent sound.

Brazilian Portuguese Examples

1. Está muito frio hoje, preciso de um cobertor. (It is very cold today, I need a blanket.)
2. Meu cobertor novo é muito fofinho. (My new blanket is very fluffy.)
3. O bebê dormiu enrolado no cobertor. (The baby slept wrapped in the blanket.)
4. Comprei um cobertor de microfibra para o inverno. (I bought a microfiber blanket for the winter.)
5. Vou colocar um cobertor extra na cama. (I am going to put an extra blanket on the bed.)

Portuguese (Portugal) Examples

1. Está muito frio hoje, preciso de uma manta. (It is very cold today, I need a blanket.)
2. A minha manta é muito macia. (My blanket is very soft.)
3. O bebé adormeceu enrolado na manta. (The baby fell asleep wrapped in the blanket.)
4. Esta manta de lã é muito quente. (This wool blanket is very warm.)
5. Vou buscar uma manta para o sofá. (I am going to get a blanket for the sofa.)