I used my brother's menorah on Chanuka , did I perform the mitzvah correctly or does the menorah have to be mine?
Various Halachic issues are at play here, in theory.
Laws of Chanukah:
You shouldn't both light in the same Menora (or Chanukiya as it's usually referred to) at the same time, unless it's very clear what night it is. (Maybe first 2 nights, no way on the 4th night.)
There's no need for the Menora to belong to the person lighting. Most of the time the Mitzva object doesn't need to belong to you, unless the Torah explicitly says so, like in Lulav where it says take for yourself.
One may not use other people's property without their permission, unless one is 100% certain they don't mind.
If one used a stolen Menora (or borrowed without permission) the Mitzva may be flawed as it comes with a sin. But it probably didn't invalidate the Mitzva, as the Menora is incidental to the Mitzva - and may not even be required. (See here for an opinion that a Menorah is required.)
If one used stolen oil/candles then possible the Mitzva is invalidated as it was done with an object causing a sin.
Assuming the simple case:
If your brother let you use his Menora while he wasn't using it, then the Mitzva was done properly.
Q: What happens if I lit someone else’s menorah by accident (can easily happen in a yeshiva)?
A: One has fulfilled his mitzvah and does not have to relight. We can rely that the owner doesn’t mind someone using his menorah for a mitzvah. (Rav Shmuel Kamenetsky Kovetzs Halachos 1:14)