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

No

While the fundamental meaning of "mora" (the third-person singular conjugation of the verb morar, meaning "to live" or "to reside") is identical in both dialects, there is a difference in typical usage regarding how people are addressed. In Brazilian Portuguese, "mora" is the standard conjugation used when addressing someone informally using the pronoun você. In Continental Portuguese, the informal "you" uses the second-person singular conjugation moras (with the pronoun tu), whereas "mora" is reserved for the third person (ele/ela) or formal address (o senhor/a senhora).

Brazilian Portuguese Examples:

1. Você mora perto daqui? (Do you live near here?)
2. Onde você mora? (Where do you live?)
3. Ela mora no Brasil. (She lives in Brazil.)
4. Ele mora sozinho. (He lives alone.)
5. Você mora com seus pais? (Do you live with your parents?)

Portuguese (Portugal) Examples:

1. Tu moras perto daqui? (Do you live near here?)
2. Onde é que tu moras? (Where do you live?)
3. Ela mora no Brasil. (She lives in Brazil.)
4. Ele mora sozinho. (He lives alone.)
5. Tu moras com os teus pais? (Do you live with your parents?)