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

No

In Brazil, colégio is a very common and broad term used to refer to private schools, covering both primary and secondary education. It is often used as a synonym for escola when the institution is not public. In Portugal, however, the word colégio has a much more specific connotation; it typically refers to a private or religious-affiliated school. To express the general idea of "school," a person from Portugal would much more naturally use escola (for primary/general education) or liceu (specifically for secondary/high school).

Brazilian Portuguese examples:

1. Meu filho estuda em um colégio particular. (My son studies at a private school.)
2. O colégio fica bem perto da minha casa. (The school is very close to my house.)
3. Preciso comprar o uniforme do colégio. (I need to buy the school uniform.)
4. As notas do colégio saíram hoje. (The school grades came out today.)
5. Ele estuda em um colégio de tempo integral. (He studies at a full-time school.)

Continental Portuguese examples:

1. O meu filho estuda numa escola privada. (My son studies at a private school.)
2. A escola fica muito perto de minha casa. (The school is very close to my house.) Note: "de minha casa" or "da minha casa" are both used, but "de minha casa" is common in PT-PT.
3. Preciso comprar o uniforme da escola. (I need to buy the school uniform.)
4. As notas da escola saíram hoje. (The school grades came out today.)
5. Ele estuda num liceu de tempo inteiro. (He studies at a high school/secondary school.)