No, it's single stage to orbit; it's a poorly written article. It goes to mach 5.4 with atmospheric air before switching to liquid oxygen. The article makes it sound like it only ever reaches mach 5.4.
It's by Jonathan Amos, who generally knows his stuff and whose articles are usually well above most bbc tech articles in quality. It's an unfortunately worded paragraph I'd say.
it is lost on me why Skylon would be noticeably better than a Mach 5-6 plane boosting a second stage. Such 2 stage system would have obviously less trade-offs and thus would be easier/cheaper to design and build.
Staging isn't free. While true that the second stage vehicle will be simpler, the combined second and first stage together could be more complicated/heavier than the Skylon.
Noticeably, the point of the Skylon and SABRE is that it's possible to use a common engine core in airbreathing and pure rocket mode, thus potentially generating weight savings.
I'm sure someone sat down and did the calculations. I have this feeling that a lot of the times when you actually sit down and work out the math for a range of assumptions, you generally come down to 'eh, maybe A, maybe B, so may as well try A'.
It's the BBC, so I'd say it's an averagely written article in their context.