Yes, they do.

It's a little different than it used to be though. Historically, the carriers require their magic signal bar algorithms/network type display by contract if you wanted to sell phones in their stores.

Now, less phones target each carrier specifically than used to, but ...

Some of them also used to require display of various network types in various situations (IE you were required to display 4G in circumstances where it was on their "4g network", regardless of whether you were using a 4g protocol)