The friendship paradox in social networks states that your friends have more friends than you do, on average. Recently, a stronger variant of the paradox was shown to hold for most people within a network: `most of your friends have more friends than you do.' Unlike the original paradox, which arises trivially because a few very popular people appear in the social circles of many others and skew their average friend popularity, the strong friendship paradox depends on features of higher-order network structures. Similar to the original paradox, the strong friendship paradox generalizes beyond popularity. When individuals have traits, many will observe that most of their friends have more of that trait than they do. This can lead to the Majority illusion, in which a rare trait will appear highly prevalent within a network. Understanding how the strong friendship paradox biases local observations within networks can inform better measurements of network structure and our understanding of collective phenomena in networks.