Don Paolo was in love with Layton, not Claire, which is why he became his nemesis.
Observe:
Don Paolo is watching Layton and Claire talk and is obviously pining. The game wants you to believe he's watching Claire, but I think he's watching Layton. After all, stalking Layton from the shadows is what he's known for.
When Claire unexpectedly kisses Layton, Don Paolo reacts with complete shock.
But when Layton smiles and returns the affection, Don Paolo completely deflates.
He is completely focused on Layton, smiling and in love. He barely glances at Claire.
And that's when he starts crying.
He doesn't acknowledge Claire at all. Instead he cries out "LAAAAAYTON!!!"
He's not mad at Layton for stealing his girl, he's mad at Layton for breaking his heart.
So he declares Layton his enemy and vows to get his revenge for making him look like a fool.
Everyone knows the very concept of a nemesis is pretty gay. As the famous saying goes:
"If a rivalry lasts longer than 7 years then you are no longer rivals. You are Gay."
Also, was anyone gonna tell me young Don Paolo wore a purple version of Luke's outfit or was I supposed to notice that by myself?