So lemme get this straight:
Lily’s meeting the MC’s sister for the first time, not knowing that the MC has already met her before since he doesn’t know where she works.
When they meet up, MC tries to disguise himself but Lily almost sees right through it - she knows it’s him but thinks it’s a phase.
The sister doesn’t recognize him, though, and begins to take a liking to him when he pats her head. Because Lily doesn’t know that the two are siblings, she now thinks of her as a rival.
So basically the introduction of the sister causes the girls to think that there’s a love triangle when there really isn’t.