Some songs just have their lyrics sung through from beginning to end twice - usually with a key change for the second time through. Is there a name for that?
I think the comment to your question that mentions 'da capo aria' as being a form in Baroque aria that plays the same piece twice with the second time embellished is interesting, but probably not exactly what you were looking for. You mention contemporary songs so it seems you're asking for a term that refers to this technique when used in contemporary music.
I can't prove the negative but I don't believe there's a standard name for that pattern used in music theory of the 20th century. While it might have been used frequently enough in Baroque times to have its own name, it's not common enough now to need a name. There might be a business term for it among music producers because, as mentioned in another comment, it's really kind of a trick for making a short song longer, and adding a little excitement by changing the key. But we'd have to ask a music producer about that.
There are a lot of variations on this pattern. You can repeat a refrain between every verse. You can repeat a whole section before going on to the next section or to the end. You can repeat melodic phrases. These patterns are usually described in terms of what they do with the music, rather than having names. There would be a lot of names to know that wouldn't get used very often.