The only cross-dressing that would make sense here is if the princess is a trans girl : no matter how much you have changed physically and grown your hair, if you don't come out, sometimes people will straight up not connect the dots. I doubt this manga will be going down this road though.
Or she's just cross dressing because men are respected more than women in that sort of environment (sexism was/is a pretty big thing), so she prefers that the outside world sees her as a male. And she's potentially a lesbian, and the easiest way to get women to come with her in that sort of time period and culture would be to make them think she's a guy at first. Nothing to do with them being trans or whatever