UK (like most of the world) uses GSM. With GSM your identity is linked to your SIM card. Put your SIM card in another phone, and "you" are still "you" as far as the phone company can tell.
Verizon (and most of US) uses a competing "standard" called CDMA, where your identity is coupled to your phone. Which means you have to involve your phone company in order to switch phone.
It's technically possible to "unlock" a CDMA phone, but in the USA, CDMA carriers refuse to unlock a phone on their network. And even if you did get an unlocked CDMA phone, the providers will refuse to activate any phones that were not originally activated on their network.
I use Verizon and they are always more than happy to activate my CDMA capable phone for use on the Verizon network - which I have been using for a very long time and live contract free this way. I also use a very old completely Unlimited plan, so buying a phone from Verizon would cancel this plan and force me on a new contract.
Anyhow my point is I have no problems adding a personal device to my plan anytime - many times.
Verizon (and most of US) uses a competing "standard" called CDMA, where your identity is coupled to your phone. Which means you have to involve your phone company in order to switch phone.