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

No.

While the word "aquecedor" exists in both dialects and refers to a device that provides heat, the answer is No because there is a difference in typical usage. In Brazil, "aquecedor" is the standard term for a water heater (aquecedor de água), whereas in Portugal, the specific term esquentador is used for that purpose. Additionally, the grammatical construction for continuous actions (the gerund in Brazil vs. "a + infinitive" in Portugal) and the preference for terms like "aquecimento central" for home heating systems also differ.

Brazilian Portuguese (Natural Usage)

1. Ligue o aquecedor, porque está muito frio. (Turn on the heater, because it is very cold.)
2. O aquecedor de água quebrou ontem. (The water heater broke yesterday.)
3. Eu estava usando o aquecedor no quarto. (I was using the heater in the bedroom.)
4. O aquecedor está fazendo um barulho estranho. (The heater is making a strange noise.)
5. Comprei um aquecedor elétrico novo. (I bought a new electric heater.)

Continental Portuguese (Natural Usage)

1. Liga o aquecedor, que está um frio! (Turn on the heater, it's freezing!)
2. O esquentador avariou esta manhã. (The water heater broke this morning.)
3. Eu estava a usar o aquecedor no quarto. (I was using the heater in the bedroom.)
4. O aquecedor está a fazer um ruído estranho. (The heater is making a strange noise.)
5. O aquecimento central da casa é antigo. (The central heating in the house is old.)