And now there's one more problem with this character. It appears in first game this is a female and in a second game it's a trap male. But nothing else about this character changed. So, I wonder if he/she needs two different IDs or not.

It's different form Hakuouki, for one it has nothing to do with the shinsengumi and it takes place sometime before the Edo period during the sekigahara battle. If you haven't played it don't assume anything.