Is the cat you adopted the same cat who once lived with another family? I mean does your cat display the same characteristics and habits he did before you took him in? Probably not. While a cat comes with certain traits—probably both inbred and learned—those traits might manifest in different ways in different home environments.
For example, a cat who is shy, might thrive in a quiet home with only one or two adult occupants. The same cat, however, in a home with children and a lot of activity and visitors, will probably quickly find a hidey spot and spend a lot of time there.
A more boisterous, active cat may become over-stimulated in a chaotic home and even develop some bad habits, such as attacking ankles, biting the hand that plays with him, and even destructive behavior—shredding paper, clawing furniture, and so forth.
If you want a quiet lap cat, make sure your home environment is suited to a docile cat. If you want to be entertained by an active clown of a cat, choose one that shows this characteristic when you visit the shelter—see if you can engage her in robust play. Then, make sure she has plenty of toys, climbing apparatus, etc to keep her stimulated and make time to play with her one-on-one.