Why when I try to assign a different text tone to one of my contacts Windows 10 mobile asks me: "Search for app in the Store?"
I'm using build: 10586.107
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As @Indrek said it works fine for me too in build 10586.494. This build is available for public.
So this must be a bug that is fixed in update. Have you tried using Preview release ring in the Insider app? This may work for you. If not there is no way to solve it as your phone cannot be updated.