This isn't 'princessy'. (unless you are letting her watch obnoxious picky princess videos, ha ha.) This is likely a sensory issue. I see this A LOT with my preschool group. I've had preschoolers which started with me who were so upset with even touching soap and wouldn't wash their hands, simply because of the texture. I have children who hate the feel of glue or paint and simply *don't* want it on their hands, so much so that they shy away from really enjoying art. The point to say, it's likely not an affectation. Some kids grow out of it sooner than others; and we all know a few adults who we'd describe as fastidious.
Here's the rub: you expect her to get messy, and she doesn't want to. So, offer her a warm, wet washcloth or a napkin at the table for her to wipe her fingers on . Teach her this aspect of self-care. She's two, so try to ignore the whining (you said she's signing, which I'm assuming means preverbal, so the whining is a distress signal) and focus on the actions; give her language "wipe hands" when you do this, so she can receive this verbal cue from you later.
Once you can see that she knows how to wipe her hands, just remind her. You'll have to do it *with* her for a while, and then, pantomime or grab your own cloth and do a little parallel play: show her how you wipe your own hands. This means that she's more likely to copy you, and continue to learn this new indendent skill.
And yes, it does get annoying at times, so try to take a few breaths and not get upset. I've been there and done that, many times!