The motorbike is clearly based on the Honda 'Monkey Bike' series of small-to-tiny motorbikes. These included (in increasing order of size) the Monkey Bike (with a tiny build, requiring the rider to squat on the narrow seat), the Gorilla (more or less the same, but with a (relatively) huge petrol tank) and the only practical bike of the three, the Honda ST or Dax, which was available at various times with 50cc - 125cc engines. As a long-term fan of the ST I still have one, and so far as I am aware they are currently available as Chinese licenced variants with modernised electrics and a 125cc engine (with a 'proper' clutch, not the jerky centrifugal variety first seen).
As for the Lunar variety, I doubt if they would have been very effective. Most off-road bikes have a high CG, making them quite bulky compared to the prototype in the picture above. I'm not at all sure how a high CG vehicle would perform off-road in 1/6th G, but my biker instincts suggest that it would have been an unhappy experience!
Finally, of course, there's the killer: dust. The LRV Grand Prix film showed us the extensive rooster tail of dust thrown up by the wheels, and Apollo 17's fender loss also showed how intolerable even spray from a rear-wheel would inevitably become. Imagine driving with your legs exposed to all that, plus the danger of spacesuit damage if you were to come off the thing.
The way forward on the Moon and Mars might include two-wheels, but if so then I think they'd need to be much more enclosed than the Monkey Bike, rather more like (I hate to say) a scooter or even the odd BMW enclosed bike from the late '90s. As for the CG issue, there's probably an answer in the form of Segway-like auto-balancers, which might make the use of two wheels (or even one) more practical (imagine a set of 'techno trousers' which the astronaut could slip into, and which would do all the balancing autonomously on an electric unicycle while keeping the rider clean - and there'd be no need to worry about penguins on the Moon or Mars!).