It’s possible that you used “Restart” rather than “Reincarnate”. It has happened to others before.
To reincarnate, you have to complete the “Cycle of Rebirth” quest, accepting the drink Gammer Wilde offers you. If you did not do this, then you may have used “Restart” instead.
As for being a Lannister again? Yes.
Some players are on their second or third play-throughs of a family so they can acquire permanent talents beyond just the fealty buildings. If you wanted, you could reincarnate 5 times in a row as a Lannister, and then switch to Stark on your 6th reincarnation.