I don't think that's true. I'm pretty sure that the acgme states it's a non-operative year that consists of 9 months of ICU training. The other 3 though are at your disposal. A lot of programs add in a month of echo, others some other electives too. If you want a month in the OR that's up to you. Some allow moonlighting, like mine, at another hospital in the system with a different Medicare number. But for the most part, it's a non-operative year.
As far as allowing a month during fellowship, what good would it do? In an accredited fellowship, you cannot be fellow and attending at the same time in the same hospital. After completing residency, do you really want to do an extra month of essentially residency just because you can't be credentialed as an attending? I'm done with that and listening to some oldies (some of whom are great, but some are awful) dictate my anesthetic. If I have a question, I'd rather run it by them and do the case myself. If you want OR time and you're in CCM or Chronic Pain, you should find a moonlighting gig as an attending.