There is no national standard.
Each state has its own (or no)
sales tax
use tax (if sales tax wasn't due / paid)
property taxes (ongoing)
and registration requirement, which may not have any relationship to the above three
and none of which pertain to federal documentation, which in principle is just titling and US residents only.
The current owner is of course responsible, and verifying the paperwork is in order is a critical part of the sales contract and transfer process, often aided by a documentation firm, deposit held in escrow etc.
For new boats moneys are collected and handled by the selling company, separate topic.
The jurisdiction can be chosen, reg / taxes payment deferred as with here, but within the framework of each jurisdiction the boat stays in in those early months after the transfer is completed.
Nothing to do with where the owner lives, like all property related taxes, where the boat is for how long is the determining factor. But some pricey boats are owned by Deleware or Wyoming etc corporations for tax avoidance.
Colorado I know you can register without the boat going near the state, like Vermont for RVs, commonly used as a transition jurisdiction by auction / charity sellers where paperwork is messy.
Some states DMV, others fish and wildlife, some county-level even local town/city level may get involved.
As I said above some states if you (the boat) stays too long, if their rate is say 14%, and you only paid 4% in the previous state, you have to pay the difference, another 10%.
Insurance may care if you are really blatantly illegal, but usually not.
Yes the taxing authority seizes your boat, so of course you need to keep all the paperwork BoS and receipts to prove that taxes are not owed, opposite to criminal burden of proof.
Fuel receipts, moorings & marinas, GPS logs, all can be relevant.
It seems like a simple national approach is needed since boats regularly move between states.
The way our federal/state system works is a mess, but bureaucrats and CPAs and lawyers are well employed and no one cares about inconveniencing "yacht" owners.
Heading overseas may simplify these issues, but of course other complications then arise.