It has to do with this difference in history:

US states: They were someone else's colonies right up until the point they were collected together into a group called the United States and only then did they stop being colonies and start self-governing.

UK internal pieces: Had a history of having been independent countries already, prior to being collected together into a group called the United Kingdom. That historical difference - that they were once self-contained countries while the US states never had such a period of history - causes the difference.

The formation of the UK was "Here's some independent countries that border each other and compete for space on this island so they have wars against each other. Instead of warring all the time maybe they can just agree to share this island by joining into one nation." (Yes, I know Northern Ireland is on a different island, but that got added to the UK later.)