Because UV radiation, particularly UVB, doesn’t cause damage by ionization, but by excitation of atoms within molecules.
Your DNA is made up of molecules that we call nucleobases (those are the A G C T and U you hear about when you’re learning about DNA or RNA). What UV light in enough intensity can do is that it will cause two neighboring T’s (thymines) to fuse together, thus rendering them unreadable by the molecules that read stuff from your genome. There’s no ionization needed there, it’s just that under certain conditions, neighboring thymines are close enough together to have some of their atoms excited and conjoin with the other thymine (specifically by exciting carbon atoms within the aromatic core of thymine)
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