I can understand "New York" or "New England" as being named to inspire hope for the future and whatnot, but why are there so many other areas that outright use the same name such as Somerset Massachusetts, Portland Maine, and Bristol Massachusetts?
Whenever a town is named for something other than physical characteristics (like 'Ford on the river Avon' is, or many US towns with Europeanised native American names), then its name reflects something symbolically important to the namer, from which we can tell what was important to them from the type of name.
In the New World, there are towns named after saints (e.g. the Christian saints like Francisco, Diego, Jose etc), and the US 'secular saints' (e.g. Jefferson, Washington, Lincoln etc), reflecting the symbolic importance of these figures to the spiritual or quasi spiritual identity of the settlers. There are towns named after concepts of importance (e.g. Providence, Liberty, Independence, Phoenix), and after locations in classical civilisation, (like Philadelphia, or Memphis), reflecting similar motivations.
Then there more prosaic names. Some are named after the owner of the town, or the sovereign (e.g. Jamestown, Charleston, Albany), which shows a political motivation, and then there are the towns named after already established locations from where the colonists came that your question is about. These cases likely reflect patriotism or pride in the origins of the settlers, as well as a link to a more familiar home in new and unfamiliar surroundings.
While I think you're right that the name New England implied the idea of a 'new and better England' in some quarters, New York was actually named this because after it was acquired by England from the Netherlands in war, it was given to James, the Duke of York, younger brother of King Charles II (and future King James II).
So while you could argue this was also to inspire hope for the future, it's important to remember that the naming reflected the expansion of the personal possessions of the Duke of York, rather than 'people from York emigrating for a better life' or anything.
Sometimes (Taunton, MA) it represented the location the settlers had come from, or (Plymouth) a port they had left: others may have followed the pattern without any particular connection to the older town. Perhaps it was seen as neutral and suggestive of loyalty to the old country when some settlers may have been suspect. The Spanish had done it too: Guadalajara, Mérida, León, presumably as a symbol of continuity with the mother country.