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

No.

While the word "trago" shares the same fundamental dictionary definitions in both variants (as the first-person singular of the verb trazer and as a noun meaning a sip or a puff), there is a difference in typical usage.

In Brazilian Portuguese, "trago" is a very common way to refer to a "sip" or "swallow" of any liquid (water, soda, beer, etc.) as well as a "drag" of a cigarette.

In Continental Portuguese, "trago" is almost exclusively used to refer to a "drag" or "puff" of a cigarette or cigar. When referring to a "sip" of a liquid, a Portuguese person will almost always use the word golo. Additionally, when translating the "same idea," note that a Portuguese person would also use sumo instead of the Brazilian suco for juice.

Brazilian Portuguese Examples:

1. "Vou dar um trago na sua água." (I'm going to take a sip of your water.)
2. "Ele deu um trago no cigarro." (He took a drag of the cigarette.)
3. "Pode dar um trago no meu suco?" (Can you take a sip of my juice?)
4. "Ele deu um trago profundo no charuto." (He took a deep puff of the cigar.)
5. "Não dei nem um trago na cerveja." (I didn't even take one sip of the beer.)

Continental Portuguese Examples:

1. "Vou dar um golo na tua água." (I'm going to take a sip of your water.)
2. "Ele deu um trago no cigarro." (He took a drag of the cigarette.)
3. "Podes dar um golo no meu sumo?" (Can you take a sip of my juice?)
4. "Ele deu um trago profundo no charuto." (He took a deep puff of the cigar.)
5. "Não dei nem um golo na cerveja." (I didn't even take one sip of the beer.)