I'm gonna try. Are you, per chance...
A Tamagotchi? Probably Nyorotchi, this one https://tamagotchi.fandom.com/wiki/Nyorotchi
I was born in the 90s.
Tamagotchis were born in the 90s
People love me even though I am full of sin.
Nyorotchi resembles a snake, which is the encarnation of sin in the holy bible. Nevertheless you still take care of him and give him food/cleaning
Committing a couple of crimes less wouldn't have changed who I am.
To evolve to Nyorotchi a tamagotchi baby needs to have plenty of failures, comitting two or three less when you're locked into Nyorotchi, does not matter, you'll get him anyways.
I cannot change even if the punishment is as severe as chopping off both my arms
Well, this is the part I cant figure, but I guess is as simple as that Nyorotchi does not have arms, actually he has a tail.