My answer: no. But I see what you are saying. It is alive in the sense that it exists, but beyond that it doesn't have any intelligence other than from the programming. I don't believe the tamagotchi inside is alive, because it has been summoned by the processor inside, and is only an avatar of/for the programming. Again, most people think tamagotchi are alive because of how they act. They are programmed to be like humans. People will give them traits because they seem so smart, but it's only the programming. That is why raising a tamagotchi is a delightful experience.
A pet dog, on the other hand, thinks for itself. It is random. It can run away if it wants to, or be very loyal to you. Tamagotchi have no choice but to respond to you when you treat them kindly. I'm going back to the intelligence thing: what fun is a pet if it can't think for itself, right? Again, that is the illusion of it having it's own mind that intrigues us.
If tamagotchi were really alive in the inside of their shell, I'd be puzzled and scared.
P.S. If you want to read a good tamagotchi fic about a tamagotchi that is 'alive', go here: http://www.mimitchi.com/html/vpet.htm (it's a suspense/mystery). I liked it very much.