I've collected license plates from a few different states but I've always had a FL drivers license. I would think if you could own a home in another state, don't see why you can't maintain other "real property" or assets. I would think the biggest hang-up would be location of primary use. If you keep it in Ohio, use it in Ohio (mostly), I would think Ohio would be happy to take your money
I could see an issue arise when you go to buy something, want to register it locally, but have an out-of-state ID. Utility bills usually solve that. Here in FL we can renew everything online.