Very much depends on the State and the bike. But, yes. We're Brits and we've done everything you want to do.
In 2001 we flew into Boston and bought two new bikes from a dealer in Brockton (about 20 miles South - the joys of the early Internet. We thought we were dealing with a Boston dealer! For various reasons, we would not recommend them.) We used an eccentric friend of my Mum's address in Brighton (suburb of Boston) to register the bike and for the insurance docs, title and AMA membership (for bike breakdown and rescue - the bikes were new, but you never know).
We put 500 miles on the clock over the first weekend and got the first service done in Brockton, then we took off up to New York State and crossed into Canada at Buffalo to see Niagara Falls. We were due to start an all brands Route 66 rally from Chicago the following week, so we hot-footed it across Ontario and crossed back into the US at Windsor. (Would not recommend this. Buying fuel in an almost deserted gas station on the outskirts of Detroit was one of the most unnerving experiences of the entire trip. If you want to cross from Canada into the US from Ontario (or vice versa) use the Sarnia border instead.
We did our Route 66 rally, made lots of friends and spent the next three months zig-zagging our way around the US, finally ending up back with (new) Canadian friends in Ontario before our final Western-bound leg.
We were due to fly out of Vancouver to start the rest of a (non-biking) RTW trip, so the intention was to sell the bikes on the West Coast of the US. We placed them 'on concession' with a dealer in Portland and caught a train back into Canada.
In fact, a year later, we were back in the UK and only one of the bikes had sold. So we arranged to ship the other - a BMW F650GS - back from California. It took months to reach us as there was a dock strike, but once we collected it from Tilbury, re-registering it in the UK was a doddle - we just had to change the headlight. Other combinations of bike manufacturer/country of import might require more work to bring them into line with national road safety standards.
ETA - I forgot to mention that my husband actually totalled his first bike in Texas. The insurance company (Arbella) were excellent and paid out enough for him to replace the bike, a Triumph Tiger 955i, with a second-hand Triumph Trophy 1200. It was the Triumph Trophy that eventually sold in Portland.