"The commoners are discriminating against me, and that's just as bad as the people in power oppressing them"
I love it when fantasy manga accidentally remind me of how much I want to eat the rich and their slimy little collaborators.
I don't think that's what she's saying at all. Your interpretation kinda shoots to the side of what she's saying rather than what she actually said, imo.
I think she's saying hating someone just because they're wealthy without any actual critique is just as terrible as hating someone just because they're poor without any actual critique. And I'd agree.
None of the nobles have a good reason for hating Kate, just as none of the commoners have a good reason for it either. Both ends of the economic spectrum see her as an upstart and "not a true [insert status here]" simply because of her parents. Kate hasn't talked down to the wealthy or poor, and Connie is being accused of drug use and partying that she hasn't done simply because of her bloodline and wealth.
If you wanna talk down to someone, let it be over something they actually did, not over what you assume of that person simply because of what they do or don't have without any actual information.
Connie is bad just because both of her parents are nobles? Kate's good because her noble father married a commoner mother?
That is, indeed, bigotry.