(Please read carefully. This isn't about sock puppets.)
Even more strangely, if you go to their network profile, you'll see two listings for Stack Overflow, but they both link to the same account, and both are listed as having the same reputation.
By the way, this isn't unique to that user; there are a total of 26 pairs of duplicate accounts tied to the same network account on Stack Overflow alone. (There were only six at the time I asked this question, but it increased to 15 in early December 2018, and as of December 2019, has ballooned to the current total of 26.) This also isn't unique to Stack Overflow; there are six pairs of duplicate accounts on Super User, five each on Server Fault and Meta Stack Exchange, and possibly more pairs on other sites as well.
There was a past case of a user having three MSE accounts tied to the same network account, but that seems to have been due to a user-specific issue, and has since been resolved. Same goes for this case from 5 years ago, and a case of a former staff member who held moderator and association bonus rights on one account but not the other, seen in earlier revisions of this post.
How was it possible for all of those users to create multiple accounts on the same site, and tie them to the same network account? Also, how can they switch between their two accounts? Can they even use the other one?