IMO the better way to design a product like this to avoid abuse would be to simply force them to sign in using each platform they want to link to. At that point

IMO the better way to design a product like this to avoid abuse would be to simply force them to sign in using each platform they want to link to. At that point all you're building is a way for people to say, "These 5 social media accounts are all me". You could allow them to select one of their accounts to source information and a photo for their combined profile. At that point you're not storing anything besides links to social media profile pages. In effect you get to piggy-back on those sites' abuse mitigation strategies (though of course you're stuck with the lowest common denominator). Your biggest decision at that point is which social media platforms to allow onto your service.