It's lovely when your cat sits on your lap, but unfortunately they usually don't. This video explains how to turn them into lap cats. Not sure how well this works, but the advice seems sensible.
At one point many moons ago my late wife and I had 7 cats, all treated the same but all individual. We had one brother and sister, the male cat was very affectionate, the female was very stand offish.
Yes, this is an incredibly huge factor. With the cats I have there are varying degrees of affection, even though they all like me very much.
One special story I have is about a ginger cat named Karel. Karel belonged to some friends (a couple) of my girlfriend of the time. They'd go on vacation fairly often and I'd usually be the one to take care of their cats, Karel and Mimie. Mimie always did her own thing, Karel on the other hand was really crazy about me. Super affectionate.
Then this one time the lady of the couple threw a birthday party. I remember it being summer, the weather was absolutely perfect, and we were all enjoying it in the garden. There were like 25 people present, all just sitting, chatting and eating cake. Karel was walking around. He passed everyone several times. He was looking for someone. Turns out he was looking specifically for me. Of all the 25 guests, Karel decided he wanted to sit on my lap, and my lap alone.
Karel, the ginger cat, will always have a special place in my heart.