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

Yes

The word "prato" is identical in meaning, spelling, and grammar in both Brazilian and European Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the "a" is typically pronounced as an open vowel [a]. In European Portuguese, the "a" is often more closed or neutralized [ɐ], and the rhythm of the word is more compressed due to the way Portuguese from Portugal tends to reduce unstressed vowels.

Brazilian Portuguese Examples

1. Eu coloquei a comida no prato. (I put the food on the plate.)
2. Esse prato está uma delícia! (This dish is delicious!)
3. Você pode lavar os pratos, por favor? (Can you wash the dishes, please?)
4. O prato principal de hoje é peixe. (The main course today is fish.)
5. Tem muita sujeira no seu prato. (There is a lot of dirt on your plate.)

European Portuguese Examples

1. Eu pus a comida no prato. (I put the food on the plate.)
2. Este prato está uma delícia! (This dish is delicious!)
3. Podes lavar a loiça, por favor? (Can you wash the dishes, please?)
4. O prato principal de hoje é peixe. (The main course today is fish.)
5. Há muita sujidade no teu prato. (There is a lot of dirt on your plate.)