Stating categorically that bamboo is stronger than steel is a bit like stating that automobiles are faster than planes. On its face, it is a bit shocking, and seems wrong. But a rocket-powered automobile is certainly going to be faster than a one-seat propeller plane over a short distance on a controlled track. Then again, the same rocket car is going to lose in a ground-speed race to an SR-71 blackbird by a factor of about 2.5. Note that each vehicle has its advantages and disadvantages aside from just measurement of speed. So it goes with broad material comparisons. Bamboo may be strong, but there are steels that will almost certainly be stronger. Bamboo fibers may be stronger than the bamboo itself, or even stronger than some steels, but will probably still not approach the strength of the strongest steels.
The highest strength steels have isotropic strength near $2\ \textrm{GPa}$, and have ductilities of $3\%$ to $10\%$. Kevlar fibers are near $3\ \textrm{GPa}$ but only along their length. Their transverse strength is closer to $50\ \textrm{MPa}$, and they have limited extensibility, closer to $1\%$. All of these values are available with some internet sleuthing and in many textbooks. Bamboo fiber data is harder to find, but the sources here and here (both links end at ScienceDirect, and are peer-reviewed journal articles) have interesting results. The former link has either $350\ \textrm{MPa}$ or $550\ \textrm{MPa}$ depending on preparation method, with no reported variance (though there is probably considerable variance, it was not reported). The latter link gives $650\pm175\ \textrm{MPa}$ and $800\pm100\ \textrm{MPa}$ for different preparation methods. Note that these values are all along the fiber lengths. No values were found for transverse strength, though it is probably an order of magnitude lower than longitudinal strength if the mechanisms are similar to those of non-biological polymers. And, at any rate, the fiber strengths are misleading since the fibers must be mixed into a polymer-matrix composite to be of any use (except perhaps as ropes), which will result in strength values lower than those of the fibers alone.
Measuring the strength of bamboo plants is challenging due to the variation in size and shape, and the difficulties of finding proper measuring tools, or even of what geometric values to measure. I was unable to find any information on the subject of bamboo plants.
As with all material selection problems, there are other tradeoffs worth considering depending on application and requirements. Some things to think about include ecological factors (bamboo is renewable), environmental factors (bamboo is flammable), loading factors (steel can be made ductile and its high properties are isotropic), and weight considerations (bamboo is much less dense). Another important factor is processing: steel is always initially made by casting to a generic shape then by working, or directly to shape by casting. The process is intensive but relatively straightforward and well understood. To form a bamboo composite one must separate the fibers from the remaining plant material, chop them to size, coat them with sizing to ensure compatibility with the matrix material, mix with matrix resin, and then finally form into shape. Some of these steps and how they affect properties are not well understood yet.
