The thing that makes something a competitor is the ability to act as a substitute. That means grocery stores that are 1000 miles away don't count. For photo sharing, what makes something a viable substitute is having a sufficient network effect, so photo sharing services with hundreds of users aren't a substitute for ones with millions.

This implies that mergers between large services that have a network effect should always be prohibited, but why is that even a problem unless your goal is to thwart competition?

It would also create a useful incentive: Federated systems (like email) have a single network that spans entities. If Microsoft wants to buy Hotmail, they're not buying a separate network so you don't have to be worried about it even if they each have 25 million users as long as that's not too large a percentage of the billion people who use email. So then companies would want to participate in federated systems instead of creating silos like modern social networks do, because then they would be as strictly prohibited from doing mergers.