Faux leather (the kind that doesn't come from an animal) feels cold and won't stretch, but genuine leather (the kind that actually comes from an animal) feels warm and will stretch. And an artificial sugar has no calories, while a genuine sugar is loaded with calories.
If you give a fake smile, you don't really feel happy. You might give a fake smile when someone is taking your picture. You're smiling because you're supposed to.
A genuine friend is a real, true friend to you: not someone who just acts like a friend sometimes. Genuine surprise is real surprise: when you're genuinely surprised, you're not just pretending. A genuine interest in something, or a genuine curiosity about something, is a real, true interest or curiosity: it's not fake or pretend. Genuine problems and genuine emergencies are real and serious: they're not silly or exaggerated.
And when you genuinely love someone, or genuinely care about something, you feel that love or care, truly and deeply, in your heart. Elsa and Anna love each other genuinely. They're sisters, so they're supposed to love each other—but they don't just fake it. Their love is genuine.