It introduced this word to us saying that it was a suit. But now it translates this word as "Outfit".
Does this refer to a fancy or professional outfit, or just any outfit? (In English suit usually designates that it is fancy.) Since suits are usually worn by men, can this be an outfit worn by a woman?
Or was it another word that was suit? Terno=suit Traje=outfit Is that correct?
Formally, yes. And also "is her(s)" and "is their(s)". To avoid confusion, we tend to prefer "dele(s)" and "dela(s)" for his/her(s)/their(s). Without a context, the sentence "o traje é seu" will be probably assumed as "the outfit is yours" because is uncommon to use it with the other meanings.
I can't listen to the slow version in this page but what you've described is pretty much what we brazilians do and I'm surprised that the machine does the same. The "o" (like the definite article masculine singular) is pronounced like English "oo" in "boot" (or the Portuguese "u") in many cases in most regions of Brazil. But when we have to speak slowly and clearly, we tend to pronounce the "o" with the "proper sound". In the given sentence, a transcription of the sounds considering brazilian average pronounciation of the alphabet would be something like "u tráji é sêu" or even with a contraction like "u tráj-é sêu". But if you ask us to speak it clearer or slower, probably you would hear something like "ô tráji é sêu" or maybe "ô trájê é sêu" (the circumflex indicates closed sounds and the other accent indicates open sounds/stress). I wish I could explain this better but it is hard to do so using a keyboard...
"Seu" is the possessive used for third person both singular and plural (ele/eles/ela/elas/você/vocês) and "teu" for the second person singular (tu). O traje é seu = The outfit is yours / his /hers / theirs). O traje é teu = The outfit is yours (singular).
Note that in Portuguese the possessive changes in number (singular or plural) according to the object possessed and not to the possesser. You would say "Os trajes são seus" even if the possesser is only one person (ele/ela/você). Does it make sense? I know it is a little bit confusing for English speakers and I'm not sure I could explain it clearly.