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

Yes

The meaning, grammar, and spelling of the word "pata" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are more open, and the final "a" is clearly and distinctly pronounced. In Continental Portuguese, there is significant vowel reduction; the final "a" is much more closed (sounding like a muffled "uh") and often becomes nearly silent in rapid speech.

Brazilian Portuguese Examples

1. O cachorro está lambendo a pata. (The dog is licking its paw.)
2. A pata do gato é bem pequena. (The cat's paw is very small.)
3. A pata está nadando no lago. (The duck is swimming in the lake.) Note: "Pata" here refers to a female duck.
4. A pata do urso é muito forte. (The bear's paw is very strong.)
5. A pata do elefante é pesada. (The elephant's paw is heavy.)

Continental Portuguese Examples

1. O cão está a lamber a pata. (The dog is licking its paw.)
2. A pata do gato é pequena. (The cat's paw is small.)
3. A pata está a nadar no lago. (The duck is swimming in the lake.)
4. A pata do urso é muito forte. (The bear's paw is very strong.)
5. A pata do leão é enorme. (The lion's paw is huge.)