I always book state government rates when available. I've been doing so for the past 10+ years with Hyatt. After hundreds of stays, I've been questioned only once about eight years ago at HR Phoenix. Fortunately, I had my state govt ID with me which they quickly glanced at. Maybe they made a note on my account and that's why I've never been asked again. I've booked a few times with the "new" govt and military business rate and also haven't been questioned. Like others, if I am traveling for work I don't get orders. I book my own accomodations and get reimbursed.
I echo the above comment about Marriott. I booked a fed rate at the Courtyard Burlington a few years ago and they analyzed my ID and said it looks like a state government agency, which doesn't qualify. I explained the per diem is the same but they were not too happy when they told me they'd make a one time exception as a courtesy.