Ok, then how do I kill it without killing the azalea?

You will need to identify and isolate the stem that is producing the different leaves and carefully remove it, roots and all. IME, just cutting back is unlikely to kill it.

I can't see any foreign stem coming from the ground, or any foreign vine intertwined with the azalea. It honestly looks like the azalea itself is sprouting foreign leaves. What am I missing?

The stem that is the focus of the last photo looks to be the euonymus....it does not have the look of an azalea stem, which tend to be thin and twiggy in appeance and start branching very low to the ground.

Perhaps better and closer photos of the entire base of the plant would help to clarify.

Figured it out. The alien stem looked exactly like an azalea branch. Very well-played by that dastardly vine. I ripped it out of the ground by the roots and separated the rest from the azalea. Fingers crossed. To azaleas fill in their own weak spots? That vine already caused a big dead spot, which I cut away.

. . . and I see your comment now. You were correct. I was fooled by the width of that stem. I expected it would be markedly different from the azalea.