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

No

While the spelling and grammar of "distrito" are identical in both varieties, there is a significant difference in typical usage and the scale of meaning.

In Brazilian Portuguese, a distrito is a small administrative subdivision of a municipality (município). It usually refers to a rural, outlying, or less urbanized area that belongs to a larger city.

In Continental Portuguese, a distrito is a large, high-level administrative division of the entire country. It is a much larger territory than the Brazilian distrito, functioning more like a province or a state (e.g., the District of Lisbon or the District of Porto).

Brazilian Portuguese Usage

1. O distrito é uma área bem tranquila da cidade. (The district is a very quiet area of the city.)
2. O prefeito visitou o distrito ontem. (The mayor visited the district yesterday.)
3. Este distrito é conhecido pela sua produção agrícola. (This district is known for its agricultural production.)
4. A estrada que leva ao distrito está em péssimas condições. (The road leading to the district is in terrible condition.)
5. O governo vai investir em saneamento para o distrito. (The government is going to invest in sanitation for the district.)

Continental Portuguese Usage

1. O distrito de Aveiro tem muita beleza natural. (The district of Aveiro has a lot of natural beauty.)
2. Eu moro no distrito de Setúbal. (I live in the district of Setúbal.)
3. O distrito de Viana do Castelo é muito bonito. (The district of Viana do Castelo is very beautiful.)
4. Ele trabalha na administração do distrito. (He works in the district administration.)
5. O distrito de Portalegre é uma região muito calma. (The district of Portalegre is a very calm region.)