Buck420 offers good advice .... these days its all about revenue income.
The dinghy rules for federally documented 'mother ships' is that such a (powered) dinghy without 'some' state registration numbers must display or be marked with a "TT mothership name" and is only to be used for direct on/off loading at a port ... not for pleasure cruising in the dinghy, etc. The TT marking means "tender to" (name of mothership).
States ignore such federal laws as this is an opportunistic revenue source for them, and require that any dinghy be registered in 'some' state ... and can be individually subject to the same state laws/rules for 'grace periods' for visiting / transient boats for 'sales tax', etc. purposes. Eg.: you only are permitted to have the dinghy AND the mothership in a state's waters for a certain amount of time ... and then you are required to pay their sales taxes or any difference between what you paid in sales taxes in your home state and the prevailing tax rate where you happen to be when you exceed that states 'grace period'. Some states have 'sojourner fees' applicable (~$1.30/ft. + registration fees for the sojourners permit) if you exceed that 'grace period', etc.
All this is contrary to federal laws for *federally documented* vessels, but states simply don't care as all they desire is 'tax revenue'. The cost to defend usually overwhelms/exceeds the cost of registration, etc. once you go beyond the time limit of individual states 'grace periods' for transient or 'visiting' vessels, etc.
BoatUS.com has such 'tax info' and listed 'grace periods' listed on their website in a 'government affairs' section. (BTW the same state tax rules apply to cars, trucks, aircraft, etc.)
Usually most states ignore dinghy tax liabilities and prefer to 'go after' the value of the mothership ... for now, but with the current situation of most state finances and deficits, it shouldnt surprise anyone that they will probably be soon going after dinghies, the value of the clothes you wear, the value of food and other personal effects, etc. that you have onboard, etc. Its all about 'tax revenue income'; and, pretty soon there will be so much tax complexity when traveling between states, that you will have to have a retained tax lawyer onboard at all times to constantly advise and defend you against aggressive 'tax piracy'.