AI/19 Entertainment own the finalists (top 12) for a year, with (I think) option on each to pay $1M to make it three years. The winner is guaranteed an album (and none of the other 11 can release an album before then), and a larger up-front payment. The others will likely be allowed to record on other labels if 19E doesn't want to, as long as 19E gets a cut.
I looked and have seen several different accounts. Some say they are obligated for a three-month tour after the show. But I suppose that does not exclude other contractual obligations. It does not seem outrageous since AI has made them all household names.