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

Yes

The word "tamanho" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference is pronunciation.

In Brazilian Portuguese, vowels are generally more "open" and the rhythm of speech is more syllable-timed (each syllable receives more equal weight). In Continental Portuguese, vowels are often "reduced" or shortened (unstressed vowels, like the 'o' at the end of the word, may sound almost like a silent "u" or a very short "schwa"), and the rhythm is more stress-timed, which can make the word sound more "clipped" to Brazilian ears.

Brazilian Portuguese Examples

1. Qual é o tamanho desse tênis? (What is the size of this sneaker?)
2. O cachorro é de tamanho grande. (The dog is large-sized.)
3. Não sei o tamanho da sua camisa. (I don't know the size of your shirt.)
4. Esse problema é de tamanho imenso. (This problem is of immense size.)
5. Essa caixa é de um tamanho estranho. (This box is of a strange size.)

Continental Portuguese Examples

1. Qual é o tamanho deste sapato? (What is the size of this shoe?)
2. O cão é de tamanho grande. (The dog is large-sized.)
3. Não sei o tamanho da tua camisa. (I don't know the size of your shirt.)
4. Esse problema é de tamanho imenso. (This problem is of immense size.)
5. Esta caixa é de um tamanho estranho. (This box is of a strange size.)