Yes you do have to register your outboard motor in OH as well as your boat & trailer. I think MO is the same. Not sure about other states? Those two states actually require a MSO (Manufacturers Statement of Origin) when a new motor is shipped. Was always a PITA for just the few states that required it! FYI.
I have gotten motors out of OH with stickers all over them. Also with them just stacked up about a 1/4" thick! Different people do things diferents ways?
Dan in TN