I've done the same trip before, and it wasn't a problem. Punctuality of the KLM early morning flight is generally quite good in my experience (unless there's severe weather or fog in the Amsterdam area, but then everything else tends to be delayed as well).
I believe that the flight time is considerably shorter than the 90 minutes scheduled, so there's some extra time baked into the schedule for late departure, taxiing from one of the remote runways to the terminal, or bussing people to the terminal. There's only one terminal complex, but you'll probably have to get from the non-Schengen part to the Schengen part, so go through passport control and security. This is usually quite fast, and there are extra lanes for short connections. 15 minutes from bus drop off to departure gate is quite doable I think.
I don't think they'd offer you the connection if it wasn't possible to make it. But there's not a lot of slack of course. But then what's the alternative? You could take the early morning flight to Frankfurt and drive down to Stuttgart, which might get you there not that much later, depending on traffic.