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

No

While the spelling and grammar of "mercado" are identical in both varieties, there is a significant difference in typical usage. In Brazil, "mercado" is a very common, everyday term used to refer to any grocery store or supermarket. In Portugal, "mercado" more specifically refers to an open-air market or a municipal market (mercado municipal); for routine grocery shopping, a Portuguese person is more likely to use "supermercado" or "mercearia" (a small local grocery shop).

Additionally, there is a difference in pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels are much more "closed" or reduced; for example, the "a" in mercado may sound like a neutral "uh" or even disappear in rapid speech, and the final "o" is much more swallowed.

Brazilian Portuguese Examples

1. Vou ao mercado comprar pão. (I'm going to the market to buy bread.)
2. O mercado está muito cheio hoje. (The market is very full today.)
3. Preciso passar no mercado antes de ir para casa. (I need to stop by the market before going home.)
4. O preço dos alimentos no mercado subiu. (The price of food at the market went up.)
5. Você pode pegar leite no mercado para mim? (Can you get milk at the market for me?)

Portuguese (Continental) Examples

1. Vou à mercearia comprar pão. (I'm going to the grocery store to buy bread.)
2. O supermercado está muito cheio hoje. (The supermarket is very full today.)
3. Tenho de passar no supermercado antes de ir para casa. (I have to stop by the supermarket before going home.)
4. O preço dos alimentos no supermercado subiu. (The price of food at the supermarket went up.)
5. Podes ir buscar leite ao supermercado para mim? (Can you go get milk at the supermarket for me?)