I'm sure they do see themselves that way, although I don't know if I'd take that as reason to think of them as such.
Your business model when your base package is free and decentralized is very different than when it is proprietary. Diaspora cannot operate like a closed startup challenging Facebook would.
In fact, and this is from my experience with building Appleseed, the main difference is in who you are trying to convince to use your software. With startups, it's users. With free software, it's administrators. Like I've written before, it's administrators choosing to set up a decentralized social networking node that will make or break Diaspora (or Appleseed or OneSocialWeb, etc).
And when your target market is administrators, any related business venture will use a B2B model, something very different than the kind of B2C startup framework we're used to putting things in.
Also, being a startup and free software project isn't mutually exclusive.