A great many companies are "bootstrapped", which means the founders kick their own money in initially (or scrounge from friends and family). Many big companies, a recent example being GitHub, were started this way.
"Have you quit your day job?"
Many people wait until they see some traction before quitting. Doesn't make it any less of a startup. Consider it a consulting gig while launching the business (unless you don't consider it a startup if people consult on the side)
"Are you seeing startup like weekly growth?"
You are falling into this paradigm trap. Not all companies see hockey stick growth initially nor do they aim to. Most sustainable businesses I can think of started slow and gradually grew up. Nothing wrong with that.
According to "The World Defined by Btipling". Most of us haven't read that book and don't care.
Have you talked to investors?
Who cares? Maybe he's self-funding it. Having outside investors isn't an absolute requirement to be a startup.
Have you quit your day job?
Who cares? Maybe he's bootstrapping the startup in his spare time while still working a dayjob. Maybe not. What f%!#ng difference does it make?
Are you seeing startup like weekly growth?
There is no such thing as "startup like weekly growth". Not all startups are going to fit the exact same model in terms of revenue growth, user growth, or whatever other metric you care to pick. And as far as that goes, he may be pre-revenue altogether. But none of that changes whether or not his thing is a "startup".
Edit: I think you also misunderstand the value proposition for your customers: you mentioned above that you should buy because you get "a full VPS, maintained by me", which is very odd.
I'm curious though: how big are you guys? I only mentioned HipChat and Campfire as they appear to be pervasive, and while I've been hearing more and more people mention Flowdock, it doesn't seem like you're an 800 pound gorilla that he should be afraid to compete against (yet!).
You could argue that he isn't competing against Hipchat, etc., but competing to convert current non-users of this kind of technology.
If they come out targeting other startups and technology companies, then I'd agree the odds are long. But come out targeting textiles manufacturers or transformer manufacturers, or shipping / trucking / railroad companies, or any of a bazillion other industries, and they may just have a shot at establishing a beach-head. Especially if they are willing to provide some vertical specific aspects to specifically appeal to customers in specific niches.