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

No

While the infinitive form "medir" shares the same meaning, spelling, and core conjugation in both Brazilian and Continental Portuguese, there is a significant difference in grammar and usage regarding how the verb is used in continuous actions. In Brazil, the gerund (medindo) is the standard way to express an ongoing action. In Portugal, the construction of the preposition "a" followed by the infinitive (a medir) is the natural way to express the same idea. Additionally, there are differences in pronunciation (Brazilian Portuguese features more open vowels and a melodic cadence, whereas Continental Portuguese features more vowel reduction) and usage of person/address (the usage of "você" in Brazil vs. "tu" in Portugal).

Brazilian Portuguese Example Sentences

1. Eu estou medindo a sala para colocar o tapete. (I am measuring the room to put the rug in.)
2. Você pode medir essa fita para mim? (Can you measure this tape for me?)
3. O pedreiro está medindo o terreno hoje. (The bricklayer is measuring the land today.)
4. A gente vai medir o espaço do sofá. (We are going to measure the space for the sofa.)
5. Eles estão medindo a altura de todo mundo. (They are measuring everyone's height.)

Continental Portuguese Example Sentences

1. Eu estou a medir a sala para colocar o tapete. (I am measuring the room to put the rug in.)
2. Podes medir esta fita por mim? (Can you measure this tape for me?)
3. O pedreiro está a medir o terreno hoje. (The bricklayer is measuring the land today.)
4. Nós vamos medir o espaço do sofá. (We are going to measure the space for the sofa.)
5. Eles estão a medir a altura de todos. (They are measuring everyone's height.)