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

No.

While the spelling and fundamental meaning of "aquecer" are the same in both dialects, there is a difference in typical usage. In Brazil, "aquecer" is often reserved for more formal, technical, or intentional heating processes (such as heating an engine, warming up muscles, or a chemical reaction). For everyday temperature changes—such as food getting hot or the weather warming up—Brazilians much more naturally use the synonym esquentar. In Portugal, "aquecer" is the standard, versatile word used for both everyday and technical contexts. Additionally, the grammatical construction for continuous actions differs, with Brazil using the gerund (aquecendo) and Portugal using the "a + infinitive" construction (a aquecer).

Brazilian Portuguese

• Você deve aquecer bem os músculos antes do treino. (You should warm up your muscles well before the workout.)
• O aquecedor serve para aquecer o quarto. (The heater is used to warm up the room.)
• Deixe o motor aquecer antes de sair com o carro. (Let the engine warm up before taking the car out.)
• O processo químico vai aquecer a solução. (The chemical process will heat up the solution.)
• O atleta precisa aquecer antes da competição. (The athlete needs to warm up before the competition.)

Continental Portuguese

• Vou aquecer o jantar agora. (I am going to heat up dinner now.)
• O dia está a aquecer bastante. (The day is getting quite warm.)
• Precisas de aquecer os músculos antes de correr. (You need to warm up your muscles before running.)
• O aquecedor vai aquecer a sala. (The heater will warm up the room.)
• O sol está a aquecer a areia da praia. (The sun is warming up the sand on the beach.)