Technologically this is really problematic. Cloning is quite difficult, nevermind twins (they have certain advantages), and our understanding of the relevant biology is not too good. Not to mention the increasingly obvious nongenetic contributions to development; "real" clones wouldn't be much like "movie" clones. It would be tricky to speed it up, too; maybe if Mendelian genetics gets connected to Darwinian evolution earlier that would help, or avoiding Lysenkoism (mainly by virtue of having a larger and more successful Soviet biological research program). Those probably aren't nearly enough, though.
Anyways, all the attention would be on people commissioning "designer" clones and the like (rich people who want themselves as heirs, basically), but the vast majority of clones would probably be dull research subjects, basically what they do already wrt stem cell research. Just take an interesting subject and clone them, then keep the cells undifferentiated. Not particularly interesting. Would-be parents already have enough combinations of IVF, surrogate mother, and sperm donor to fulfill any desire for children, so the only people using this would be the ones who want a child very close to a particular person--ie., the designer clone people--or the people who want cell lines as similar as possible to each other for research.
Cloning people for organ extraction is unlikely (cloning people to use the resulting stem cells to build a new organ is more likely), given the enormous ethical problems. Similarly, so would building a massive set of clones for some kind of long-term research project--the resulting data would be both amazing and amazingly unethical to obtain.