That's very fuzzy. What about e-mail? Or the phone network? Or the internet in general? Yes, there can be some social aspects to it. But that by itself does not make it into a social network.

Networks have topologies and paths. The social graph matters on facebook because you get connected to your friend's friends which is a core feature of the platform. This is not the case with chat platforms.

It is only a chat platform and it predates the first social networks (which were explicitly called "social networks" back then). This is a bit of a pet peeve of mine, but I think people use the term "social network" way to much so that it doesn't even mean anything any more.

We associate other stuff like sharing and virality with social networks. Isn't the phone network just as much of a social network as ICQ?

One distinct quality is that as a user you actually have the possibility to navigate the network. On ICQ you only see your own contact list. You cannot see how many friends/contacts/followers/relations anyone else has. You cannot traverse the network from your direct friends to indirect friends.

A social network puts the connections you have with other people prominently on display for everyone to see. The number of "friends" you have is a central social status metric on social networks.